{
  "metadata": {
    "success": true,
    "status_code": 200,
    "request": {
      "request_id": "req-mat-subtract-020e65edbd884eae",
      "correlation_id": "cor-mat-subtract-4c9bb5974cba4ff6",
      "processing_time_ms": 124,
      "timestamp_iso": "2026-08-06T15:29:14.695773Z",
      "input_hash": "mat-subtract-3712ea679c8b95c8"
    },
    "usage": {
      "plan": "free",
      "limit": 20,
      "requests_used": 1,
      "remaining": 19,
      "resets_at": "2026-08-07T15:29:13.811788Z"
    },
    "api": {
      "level": "detailed",
      "version": "0.1.0",
      "schema_version": "0.1.0",
      "deprecation": {
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      }
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      "compression_ratio": 4.58,
      "compression_algorithm": "gzip"
    }
  },
  "taxonomy": {
    "domain": "Mathematics",
    "branch": "Linear Algebra",
    "module": "Matrices",
    "topic": "Arithmetic",
    "concept": "Matrix Subtraction",
    "concept_slug": "subtract"
  },
  "endpoint": {
    "method": "POST",
    "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/subtract",
    "description": "Compute the difference of two matrices."
  },
  "task": [
    "Compute the difference of the matrices \\( A \\) and \\( {\\color{blue}{B}} \\) given below.",
    "\\[ A = \\left[\\begin{array}{rrr} 5 & 1 & 6 \\\\ 3 & 4 & 5 \\\\ 6 & 4 & 8 \\end{array}\\right] \\:, \\quad {\\color{blue}{B}} = {\\color{blue}{\\left[\\begin{array}{rrr} -4 & -2 & 6 \\\\ 4 & 7 & -3 \\\\ 7 & -3 & 5 \\end{array}\\right]}} \\]"
  ],
  "normalized_request": {
    "tab_main": {
      "name": "tab_main",
      "label": "Main tab",
      "type": "str",
      "description": null,
      "choices": null,
      "value": null,
      "latex": null,
      "category": "group"
    },
    "a": {
      "name": "a",
      "label": "Matrix A",
      "type": "matrix",
      "description": "The first input matrix.",
      "choices": null,
      "value": [
        [
          5,
          1,
          6
        ],
        [
          3,
          4,
          5
        ],
        [
          6,
          4,
          8
        ]
      ],
      "latex": "\\( \\left[\\begin{array}{rrr} 5 & 1 & 6 \\\\ 3 & 4 & 5 \\\\ 6 & 4 & 8 \\end{array}\\right] \\)",
      "category": "input"
    },
    "b": {
      "name": "b",
      "label": "Matrix B",
      "type": "matrix",
      "description": "The second input matrix or scalar. If it is a matrix, it must have the same size as that of Matrix A.",
      "choices": null,
      "value": [
        [
          -4,
          -2,
          6
        ],
        [
          4,
          7,
          -3
        ],
        [
          7,
          -3,
          5
        ]
      ],
      "latex": "\\( \\left[\\begin{array}{rrr} -4 & -2 & 6 \\\\ 4 & 7 & -3 \\\\ 7 & -3 & 5 \\end{array}\\right] \\)",
      "category": "input"
    },
    "tab_options": {
      "name": "tab_options",
      "label": "Options tab",
      "type": "str",
      "description": null,
      "choices": null,
      "value": null,
      "latex": null,
      "category": "group"
    },
    "response_profile": {
      "name": "response_profile",
      "label": "Response Profile",
      "type": "str",
      "description": "Controls results that are returned.",
      "choices": {
        "lite": "Returns metadata, taxonomy and answer.",
        "standard": "Returns metadata, taxonomy, answer and steps.",
        "detailed": "Complete API response including: metadata, taxonomy, task, normalized_request, intermediate, result, steps, quick_actions, explore, learn, implementations, exercises, and notes."
      },
      "value": "detailed",
      "latex": null,
      "category": "options"
    },
    "evaluate_expressions": {
      "name": "evaluate_expressions",
      "label": "Evaluate Expressions",
      "type": "bool",
      "description": "Whether or not to evaluate (simplify) expressions.",
      "choices": null,
      "value": true,
      "latex": null,
      "category": "options"
    },
    "apply_decimals_in_steps": {
      "name": "apply_decimals_in_steps",
      "label": "Apply Decimals in Steps",
      "type": "bool",
      "description": "If `True`, intermediate calculations in step-by-step result will be rounded according to the `decimals` setting, potentially introducing approximations. If `False`, exact values will be used in all intermediate steps. Discarded if `decimals=None` or `decimals=-1`.",
      "choices": null,
      "value": false,
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      "category": "options"
    },
    "include": {
      "name": "include",
      "label": "Included Sections",
      "type": "vector_str",
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      "choices": null,
      "value": null,
      "latex": null,
      "category": "input"
    },
    "footer": {
      "name": "footer",
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      "type": "str",
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      "choices": null,
      "value": null,
      "latex": null,
      "category": "group"
    },
    "result_name": {
      "name": "result_name",
      "label": "Result Name",
      "type": "str",
      "description": "The designated name for the final computed output.",
      "choices": null,
      "value": "M",
      "latex": null,
      "category": "input"
    },
    "steps_bg": {
      "name": "steps_bg",
      "label": "Background",
      "type": "bool",
      "description": "Indicates whether background theory should be provided alongside the solution.",
      "choices": null,
      "value": true,
      "latex": null,
      "category": "options"
    },
    "decimals": {
      "name": "decimals",
      "label": "Decimals",
      "type": "int",
      "description": "Number of decimal places (or significant figures) to round the results (None or -1 means no rounding).",
      "choices": null,
      "value": 5,
      "latex": null,
      "category": "input"
    }
  },
  "intermediate": null,
  "result": {
    "name": "M",
    "answer": [
      [
        9,
        3,
        0
      ],
      [
        -1,
        -3,
        8
      ],
      [
        -1,
        7,
        3
      ]
    ],
    "properties": {
      "kind": "matrix",
      "num_variables": null,
      "shape": [
        3,
        3
      ],
      "description": "3 x 3 matrix",
      "size_bytes": 72,
      "labels": null
    },
    "representations": {
      "plain_text": "[[9, 3, 0], [-1, -3, 8], [-1, 7, 3]]",
      "latex": "\\( {\\color{blue}{M = \\left[\\begin{array}{rrr} 9 & 3 & 0 \\\\ -1 & -3 & 8 \\\\ -1 & 7 & 3 \\end{array}\\right]}} \\)",
      "csv": "9,3,0\r\n-1,-3,8\r\n-1,7,3"
    }
  },
  "steps": [
    {
      "id": "step-1-background",
      "content": "STEP 1: Background",
      "content_type": "h1",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-BG-001",
      "content": "Let \\( A \\) and \\( {\\color{blue}{B}} \\) be two matrices of the same order \\( m \\times n \\).",
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-BG-002",
      "content": "\\[ A = \\left[\\begin{array}{cccc} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{array} \\right] \\:,\\quad {\\color{blue}{B = \\left[\\begin{array}{cccc} b_{11} & b_{12} & \\cdots & b_{1n} \\\\ b_{21} & b_{22} & \\cdots & b_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ b_{m1} & b_{m2} & \\cdots & b_{mn} \\end{array} \\right]}} \\]",
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-BG-003",
      "content": "The matrix \\( C \\) is obtained by subtracting the corresponding entries of \\( {\\color{blue}{B}} \\) from those of \\( A \\).",
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-BG-004",
      "content": "\\[C = \\left[\\begin{array}{cccc} a_{11} - {\\color{blue}{b_{11}}} & a_{12} - {\\color{blue}{b_{12}}} & \\cdots & a_{1n} - {\\color{blue}{b_{1n}}} \\\\ a_{21} - {\\color{blue}{b_{21}}} & a_{22} - {\\color{blue}{b_{22}}} & \\cdots & a_{2n} - {\\color{blue}{b_{2n}}} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} - {\\color{blue}{b_{m1}}} & a_{m2} - {\\color{blue}{b_{m2}}} & \\cdots & a_{mn} - {\\color{blue}{b_{mn}}} \\end{array} \\right] \\]",
      "content_type": "latex",
      "metadata": null,
      "explain": "Each entry \\( c_{ij} \\) of matrix \\( C \\) is computed as \\( c_{ij} = a_{ij} - {\\color{blue}{b_{ij}}} \\).",
      "explain_type": "info"
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-RF-001",
      "content": "",
      "content_type": "border",
      "metadata": {
        "width": 1,
        "pattern": "solid",
        "color": "skyblue"
      },
      "explain": null,
      "explain_type": null
    },
    {
      "id": "step-2-the-input-matrices",
      "content": "STEP 2: The Input Matrices",
      "content_type": "h1",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-001",
      "content": "Write down the two matrices.",
      "content_type": "text",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-002",
      "content": [
        "\\( \\quad A = \\left[\\begin{array}{rrr} 5 & 1 & 6 \\\\ 3 & 4 & 5 \\\\ 6 & 4 & 8 \\end{array}\\right] \\:, \\quad {\\color{blue}{B}} \\: {\\color{blue}{=}} \\: {\\color{blue}{\\left[\\begin{array}{rrr} -4 & -2 & 6 \\\\ 4 & 7 & -3 \\\\ 7 & -3 & 5 \\end{array}\\right]}} \\)"
      ],
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "step-3-subtract-the-two-matrices",
      "content": "STEP 3: Subtract the Two Matrices",
      "content_type": "h1",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-003",
      "content": "\\( M = A - {\\color{blue}{B}} \\)",
      "content_type": "latex",
      "metadata": null,
      "explain": "Subtract each entry of \\( {\\color{blue}{B}} \\) from the corresponding entry of \\( A \\) element-wise. That is \\( A - B = [a_{ij} - b_{ij}] \\).",
      "explain_type": "info"
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-004",
      "content": "\\( \\quad = \\left[\\begin{array}{rrr} 5 - \\left({\\color{blue}{-4}}\\right) & 1 - \\left({\\color{blue}{-2}}\\right) & 6 - {\\color{blue}{6}} \\\\ 3 - {\\color{blue}{4}} & 4 - {\\color{blue}{7}} & 5 - \\left({\\color{blue}{-3}}\\right) \\\\ 6 - {\\color{blue}{7}} & 4 - \\left({\\color{blue}{-3}}\\right) & 8 - {\\color{blue}{5}} \\end{array}\\right] \\)",
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "step-4-calculate-each-entry",
      "content": "STEP 4: Calculate Each Entry",
      "content_type": "h1",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-005",
      "content": "Calculate each element-wise difference shown above to get the final answer.",
      "content_type": "text",
      "metadata": null,
      "explain": null,
      "explain_type": null
    },
    {
      "id": "MATH-LALG-MATRIX-SUBTRACT-006",
      "content": "\\( {\\color{blue}{M = \\left[\\begin{array}{rrr} 9 & 3 & 0 \\\\ -1 & -3 & 8 \\\\ -1 & 7 & 3 \\end{array}\\right]}}{\\color{red}{\\checkmark}}{\\color{red}{\\checkmark}} \\)",
      "content_type": "latex",
      "metadata": null,
      "explain": null,
      "explain_type": null
    }
  ],
  "quick_actions": [
    {
      "label": "Determinant 2 x 2 matrix",
      "api_route": "/v1/mathematics/linear-algebra/matrices/determinant-2x2",
      "description": "Compute the determinant of a 2x2 matrix."
    },
    {
      "label": "Determinant 3 x 3 matrix - Sarrus",
      "api_route": "/v1/mathematics/linear-algebra/matrices/determinant-3x3",
      "description": "Compute the determinant of a 3x3 matrix using the Sarrus rule."
    },
    {
      "label": "Determinant n x n matrix - Gauss",
      "api_route": "/v1/mathematics/linear-algebra/matrices/determinant-nxn",
      "description": "Compute the determinant of an nxn matrix using the Gaussian elimination method."
    }
  ],
  "explore": {
    "Dimensions": {
      "nrows": {
        "label": "Number of rows",
        "api_route": "/v1/mathematics/linear-algebra/matrices/dimensions/nrows",
        "description": ""
      },
      "ncols": {
        "label": "Number of columns",
        "api_route": "/v1/mathematics/linear-algebra/matrices/dimensions/ncols",
        "description": ""
      },
      "shape": {
        "label": "Shape (rows, columns)",
        "api_route": "/v1/mathematics/linear-algebra/matrices/dimensions/shape",
        "description": ""
      }
    },
    "Minimum": {
      "row-min": {
        "label": "Row minimum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-min",
        "description": ""
      },
      "col-min": {
        "label": "Column minimum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-min",
        "description": ""
      },
      "mat-min": {
        "label": "Matrix minimum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-min",
        "description": ""
      }
    },
    "Maximum": {
      "row-max": {
        "label": "Row maximum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-max",
        "description": ""
      },
      "col-max": {
        "label": "Column maximum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-max",
        "description": ""
      },
      "mat-max": {
        "label": "Matrix maximum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-max",
        "description": ""
      }
    },
    "Sum": {
      "row-sum": {
        "label": "Row sum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-sum",
        "description": ""
      },
      "col-sum": {
        "label": "Column sum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-sum",
        "description": ""
      },
      "mat-sum": {
        "label": "Matrix sum",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-sum",
        "description": ""
      }
    },
    "Mean": {
      "row-mean": {
        "label": "Row mean",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-mean",
        "description": ""
      },
      "col-mean": {
        "label": "Column mean",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-mean",
        "description": ""
      },
      "mat-mean": {
        "label": "Matrix mean",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-mean",
        "description": ""
      }
    },
    "Variance": {
      "row-var": {
        "label": "Row variance",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-var",
        "description": ""
      },
      "col-var": {
        "label": "Column variance",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-var",
        "description": ""
      },
      "mat-var": {
        "label": "Matrix variance",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-var",
        "description": ""
      }
    },
    "Standard deviation": {
      "row-std": {
        "label": "Row standard deviation",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/row-std",
        "description": ""
      },
      "col-std": {
        "label": "Column standard deviation",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/col-std",
        "description": ""
      },
      "mat-std": {
        "label": "Matrix standard deviation",
        "api_route": "/v1/mathematics/linear-algebra/statistics/descriptive/mat-std",
        "description": ""
      }
    },
    "Index & Slice": {
      "index-and-slice": {
        "label": "Index and slice",
        "api_route": "/v1/mathematics/linear-algebra/create/index-and-slice",
        "description": ""
      }
    },
    "Insert": {
      "row-insert": {
        "label": "Insert rows",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/row-insert",
        "description": ""
      },
      "col-insert": {
        "label": "Insert columns",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/col-insert",
        "description": ""
      }
    },
    "Concatenate": {
      "row-concat": {
        "label": "Concatenate along rows",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/row-concat",
        "description": ""
      },
      "col-concat": {
        "label": "Concatenate along columns",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/col-concat",
        "description": ""
      }
    },
    "Delete": {
      "row-delete": {
        "label": "Delete rows",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/row-delete",
        "description": ""
      },
      "col-delete": {
        "label": "Delete columns",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/col-delete",
        "description": ""
      }
    },
    "Transpose & Triangular": {
      "transpose": {
        "label": "Transpose",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/transpose",
        "description": ""
      },
      "tril": {
        "label": "Lower triangular matrix",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/tril",
        "description": ""
      },
      "triu": {
        "label": "Upper triangular matrix",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/triu",
        "description": ""
      }
    },
    "Flip & Rotate": {
      "flipud": {
        "label": "Flip rows (up/down)",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/flipud",
        "description": ""
      },
      "fliplr": {
        "label": "Flip columns (left/right)",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/fliplr",
        "description": ""
      },
      "rot90": {
        "label": "Rotate matrix",
        "api_route": "/v1/mathematics/linear-algebra/change-appearance/rot90",
        "description": ""
      }
    },
    "Square Matrix Operations": {
      "trace": {
        "label": "Trace",
        "api_route": "/v1/mathematics/linear-algebra/matrices/trace",
        "description": ""
      },
      "determinant-3x3": {
        "label": "Determinant",
        "api_route": "/v1/mathematics/linear-algebra/matrices/determinant-3x3",
        "description": ""
      },
      "inverse-3x3": {
        "label": "Inverse",
        "api_route": "/v1/mathematics/linear-algebra/matrices/inverse-3x3",
        "description": ""
      },
      "charpoly": {
        "label": "Characteristic polynomial",
        "api_route": "/v1/mathematics/linear-algebra/create/charpoly",
        "description": ""
      }
    },
    "Matrix Forms": {
      "ref": {
        "label": "Row echelon form",
        "api_route": "/v1/mathematics/linear-algebra/matrix-forms/ref",
        "description": ""
      },
      "rref": {
        "label": "Reduced row echelon form",
        "api_route": "/v1/mathematics/linear-algebra/matrix-forms/rref",
        "description": ""
      }
    }
  },
  "learn": {
    "education_level": {
      "level": 3,
      "label": "alevel",
      "display_name": "A-Level",
      "description": null,
      "age_range": "16-19"
    },
    "summary": "Matrix addition is the operation of combining two matrices by adding their corresponding entries. It is defined only for matrices of the same order, meaning they have the same number of rows and columns. The resulting matrix has the same dimensions as the original matrices, with each entry equal to the sum of the entries in the corresponding positions. Matrix addition satisfies important properties such as commutativity and associativity, making it a fundamental operation in linear algebra and an essential tool for combining mathematical models and matrix representations.",
    "prerequisites": [
      "Understanding of matrices and their dimensions (order).",
      "Knowledge of rows, columns, and matrix entries.",
      "Ability to perform addition of real numbers and algebraic expressions.",
      "Understanding of matrix notation.",
      "Ability to determine whether two matrices are conformable for addition."
    ],
    "intuition": [
      "Matrix addition extends the familiar process of adding numbers to rectangular arrays by combining entries that occupy the same position in each matrix.",
      "The requirement that two matrices have the same order ensures that every entry in one matrix corresponds to exactly one entry in the other matrix.",
      "Matrix addition changes only the values of corresponding entries, not the structure of the matrices, so the resulting matrix has the same dimensions as the original matrices.",
      "Matrix addition provides a natural way to combine matrices that represent related quantities, datasets, or mathematical models with identical dimensions."
    ],
    "concepts": [
      "Definition of matrix addition.",
      "Matrix order (dimensions).",
      "Compatibility of matrices for addition.",
      "Entrywise addition of corresponding entries.",
      "Commutative property of matrix addition.",
      "Associative property of matrix addition.",
      "The zero matrix as the additive identity.",
      "Applications of matrix addition."
    ],
    "key_terms": [
      "Matrix addition",
      "Matrix",
      "Matrix notation",
      "Matrix order",
      "Matrix dimensions",
      "Compatible matrices",
      "Rows",
      "Columns",
      "Matrix entries",
      "Corresponding entries",
      "Entrywise addition",
      "Zero matrix",
      "Additive identity",
      "Commutative property",
      "Associative property"
    ],
    "formulae": [
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-001",
        "name": "Matrix addition definition",
        "latex": "\\( A + B = [a_{ij} + b_{ij}] \\)",
        "description": "If \\( A = [a_{ij}] \\) and \\(B = [b_{ij}] \\) are matrices of the same order \\( m \\times n \\), then their sum is obtained by adding corresponding entries.",
        "conditions": [
          "\\( A \\) and \\( B \\) must have the same order \\( m \\times n \\)."
        ],
        "variables": [
          "\\( A = [a_{ij}] \\) - First matrix",
          "\\( B = [b_{ij}] \\) - Second matrix",
          "\\( a_{ij} \\) - Entry of matrix \\( A \\) at row \\( i \\), column \\( j \\)",
          "\\( b_{ij} \\) - Entry of matrix \\( B \\) at row \\( i \\), column \\( j \\)",
          "\\( m \\) - Number of rows in matrices \\( A \\) and \\( B \\)",
          "\\( n \\) - Number of columns in matrices \\( A \\) and \\( B \\)"
        ],
        "formula_type": "definition"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-002",
        "name": "Element-wise matrix addition",
        "latex": "\\( (A + B)_{ij} = a_{ij} + b_{ij} \\)",
        "description": "The \\( (i, j) \\)-th entry of the resulting matrix is obtained by adding the entries in the same position from \\( A \\) and \\( B \\).",
        "conditions": [
          "\\( i = 1, 2, \\ldots, m \\)",
          "\\( j = 1, 2, \\ldots, n \\)"
        ],
        "variables": [
          "\\( i \\) - Row index, where \\( i \\) ranges from 1 to \\( m \\)",
          "\\( j \\) - Column index, where \\( j \\) ranges from 1 to \\( n \\)",
          "\\( a_{ij} \\) - Entry in the \\( i \\)-th row and \\( j \\)-th column of matrix \\( A \\)",
          "\\( b_{ij} \\) - Entry in the \\( i \\)-th row and \\( j \\)-th column of matrix \\( B \\)"
        ],
        "formula_type": "rule"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-003",
        "name": "Dimension preservation",
        "latex": "\\( A, B \\in \\mathbb{R}^{m\\times n}\\Rightarrow A + B\\in\\mathbb{R}^{m\\times n} \\)",
        "description": "The resulting matrix has the same dimensions as the original matrices.",
        "conditions": [
          "Input matrices must have identical dimensions."
        ],
        "variables": [
          "\\( A \\) - First matrix, must have same dimensions as \\( B \\)",
          "\\( B \\) - Second matrix, must have same dimensions as \\( A \\)",
          "\\( m \\) - Number of rows (must match for both matrices)",
          "\\( n \\) - Number of columns (must match for both matrices)"
        ],
        "formula_type": "property"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-004",
        "name": "Commutative property of matrix addition",
        "latex": "\\( A + B = B + A \\)",
        "description": "The order of addition does not affect the result.",
        "conditions": [
          "\\( A \\) and \\( B \\) must have the same dimensions."
        ],
        "variables": [
          "\\( A \\) - First matrix, same dimensions as \\( B \\)",
          "\\( B \\) - Second matrix, same dimensions as \\( A \\)"
        ],
        "formula_type": "property"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-005",
        "name": "Associative property of matrix addition",
        "latex": "\\( (A + B) + C = A + (B + C) \\)",
        "description": "The grouping of matrices during addition does not affect the result.",
        "conditions": [
          "\\( A \\), \\( B \\), and \\( C \\) must have compatible dimensions."
        ],
        "variables": [
          "\\( A \\) - First matrix, must have same dimensions as \\( B \\) and \\( C \\)",
          "\\( B \\) - Second matrix, must have same dimensions as \\( A \\) and \\( C \\)",
          "\\( C \\) - Third matrix, must have same dimensions as \\( A \\) and \\( B \\)"
        ],
        "formula_type": "property"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-FRM-006",
        "name": "Additive identity property",
        "latex": "\\( A + 0 = A \\)",
        "description": "Adding the zero matrix of the same order as \\( A \\) leaves the matrix unchanged.",
        "conditions": [
          "The zero matrix must have the same dimensions as \\( A \\)."
        ],
        "variables": [
          "\\( A \\) - Any \\( m \\times n \\) matrix being added to the zero matrix"
        ],
        "formula_type": "identity"
      }
    ],
    "objectives": [
      "Define matrix addition and explain the conditions required for two matrices to be added.",
      "Identify the order, dimensions, and corresponding entries of matrices involved in an addition operation.",
      "Determine whether two or more matrices are compatible for addition based on their dimensions.",
      "Perform matrix addition by calculating the sums of corresponding entries.",
      "Express matrix addition using standard mathematical notation and entrywise formulas.",
      "Apply the properties of matrix addition, including the commutative property, associative property, and additive identity.",
      "Solve problems involving matrix addition in contexts such as data representation, mathematical models, and applied linear algebra."
    ],
    "hints": [
      "Check that the matrices have the same order before attempting addition.",
      "Add only corresponding entries that occupy the same row and column position.",
      "The resulting matrix keeps the same dimensions as the original matrices.",
      "Match each row and column carefully when combining corresponding entries.",
      "Matrix addition applies the same arithmetic rules as scalar addition to each pair of corresponding entries.",
      "If two matrices have different dimensions, matrix addition is not defined."
    ],
    "applications": [
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-001",
        "title": "Combining Data Matrices",
        "industry": "Data Science",
        "description": "Combining datasets represented as matrices by adding corresponding entries from different sources or measurements."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-002",
        "title": "Economic Data Aggregation",
        "industry": "Economics",
        "description": "Aggregating economic data such as sales, production, or financial records when values share the same matrix structure."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-003",
        "title": "Numerical Model Updates",
        "industry": "Engineering",
        "description": "Updating numerical models by combining matrices that represent related systems, measurements, or physical processes."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-004",
        "title": "Feature Matrix Combination",
        "industry": "Machine Learning",
        "description": "Combining feature matrices, model parameters, or numerical representations used in computational learning systems."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-005",
        "title": "Image Matrix Processing",
        "industry": "Image Processing",
        "description": "Combining pixel matrices to modify images, adjust brightness, or process numerical image representations."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-APP-006",
        "title": "Statistical Data Organization",
        "industry": "Statistics",
        "description": "Supporting matrix-based calculations for organizing, combining, and analyzing structured statistical data."
      }
    ],
    "real_world_examples": [
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-001",
        "title": "Combining Regional Sales Data",
        "industry": "Business Analytics",
        "description": "A company combines monthly sales data from two regions by adding corresponding entries in their sales matrices to obtain total sales for each product category and month."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-002",
        "title": "Combining Experimental Measurements",
        "industry": "Scientific Research",
        "description": "Researchers combine measurements from multiple experiments where observations are organized in identical matrix formats, allowing corresponding measurements to be added and analyzed."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-003",
        "title": "Combining Image Intensity Matrices",
        "industry": "Image Processing",
        "description": "A graphics system combines grayscale image data by adding corresponding pixel intensity matrices to adjust brightness or merge image information."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-004",
        "title": "Aggregating Financial Records",
        "industry": "Finance",
        "description": "A company aggregates financial records by adding expense or revenue matrices from different departments where rows and columns represent matching categories and time periods."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-005",
        "title": "Combining Sensor Measurements",
        "industry": "Sensor Systems",
        "description": "An environmental monitoring system combines measurements from multiple sensors by adding matrices where each entry represents a measurement recorded at the same location and time."
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-EX-006",
        "title": "Merging Geographic Datasets",
        "industry": "Geographic Information Systems",
        "description": "A mapping system merges transportation, population, or geographic datasets represented as matrices with identical dimensions by adding corresponding entries."
      }
    ],
    "misconceptions": [
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-001",
        "category": "definition_error",
        "misconception": "Assuming that any two matrices can be added without checking that they have the same order.",
        "correct": "Two matrices can only be added when they have the same order and dimensions.",
        "why_wrong": "Matrix addition requires corresponding entries \\( a_{ij} \\) and \\( b_{ij} \\) to exist in both matrices.",
        "example": {
          "incorrect": "\\( A_{2 \\times 3} + B_{3 \\times 2} \\)",
          "correct": "\\( A_{2 \\times 3} + B_{2 \\times 3} \\)"
        }
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-002",
        "category": "procedure_error",
        "misconception": "Adding matrices by combining rows or columns instead of adding corresponding entries.",
        "correct": "Matrix addition is performed entrywise using entries with the same row and column indices.",
        "why_wrong": "The entry \\( c_{ij} \\) depends only on \\( a_{ij} \\) and \\( b_{ij} \\), not on entire rows or columns.",
        "example": {
          "incorrect": "Adding entire rows: \\( R_1(A) + R_1(B) \\)",
          "correct": "\\( c_{11} = a_{11} + b_{11}, \\ c_{12} = a_{12} + b_{12} \\)"
        }
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-003",
        "category": "conceptual_error",
        "misconception": "Believing that matrix addition changes the dimensions of the matrices being added.",
        "correct": "The resulting matrix has the same order as the matrices being added.",
        "why_wrong": "Matrix addition changes entries but does not change the matrix structure.",
        "example": {
          "incorrect": "\\( 3 \\times 3 + 3 \\times 3 = 6 \\times6 \\)",
          "correct": "\\( 3 \\times 3 + 3 \\times 3 = 3 \\times 3 \\)"
        }
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-004",
        "category": "conceptual_error",
        "misconception": "Confusing matrix addition with matrix multiplication.",
        "correct": "Matrix addition is performed entrywise, while multiplication uses row-column operations.",
        "why_wrong": "The operations have different definitions and compatibility requirements.",
        "example": {
          "incorrect": "\\( A + B \\) computed using row-column products",
          "correct": "\\( \\displaystyle (AB)_{ij} = \\sum_k a_{ik} b_{kj} \\)"
        }
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-005",
        "category": "definition_error",
        "misconception": "Assuming matrices of different sizes can be added by treating missing entries as zero.",
        "correct": "Matrix addition is undefined for matrices with different orders.",
        "why_wrong": "Missing entries are not automatically replaced with zeros.",
        "example": {
          "incorrect": "\\( A_{2 \\times 3} + B_{2 \\times 2} \\)",
          "correct": "\\(A_{2 \\times 3} + B_{2 \\times 3} \\)"
        }
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-MC-006",
        "category": "property_error",
        "misconception": "Forgetting that the zero matrix must have the same order as the matrix it is added to.",
        "correct": "The zero matrix must have the same dimensions as the original matrix.",
        "why_wrong": "The additive identity property requires matching dimensions.",
        "example": {
          "incorrect": "\\( A_{2 \\times 3} + 0_{2 \\times 2} \\)",
          "correct": "\\( A_{2 \\times 3} + 0_{2 \\times 3} = A_{2 \\times 3} \\)"
        }
      }
    ],
    "estimated_solve_time": 30,
    "recommended_next_topics": [
      {
        "id": "MATH-LALG-MATRIX-ADD-TOP-001",
        "concept": "Matrix Subtraction",
        "relationship_type": "next_topic",
        "description": "TODOTODOTODO",
        "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/subtract"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-TOP-002",
        "concept": "Element-wise Matrix Multiplication",
        "relationship_type": "extension",
        "description": "TODOTODOTODO",
        "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/multiply-ew"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-TOP-003",
        "concept": "Element-wise Matrix Division",
        "relationship_type": "extension",
        "description": "TODOTODOTODO",
        "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/divide-ew"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-TOP-004",
        "concept": "Element-wise Matrix Power",
        "relationship_type": "extension",
        "description": "TODOTODOTODO",
        "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/power-ew"
      },
      {
        "id": "MATH-LALG-MATRIX-ADD-TOP-005",
        "concept": "Standard Matrix Multiplication",
        "relationship_type": "extension",
        "description": "TODOTODOTODO",
        "route": "/v1/mathematics/linear-algebra/matrices/arithmetic/matmul"
      }
    ],
    "difficulty": {
      "level": 2,
      "label": "easy",
      "display_name": "Easy",
      "description": "Requires fundamental knowledge and direct application.",
      "age_range": null
    },
    "skill_level": {
      "level": 2,
      "label": "elementary",
      "display_name": "Elementary",
      "description": "Learners with basic foundational skills.",
      "age_range": null
    },
    "bloom_level": {
      "level": 3,
      "label": "apply",
      "display_name": "Apply",
      "description": "Use knowledge to solve problems.",
      "age_range": null
    },
    "tags": [
      "mathematics",
      "linear_algebra",
      "matrices",
      "matrix_operations",
      "matrix_addition",
      "matrix_arithmetic",
      "matrix_order",
      "matrix_dimensions",
      "matrix_entries",
      "corresponding_entries",
      "entrywise_operations",
      "additive_identity"
    ],
    "references": [
      {
        "id": "MATH-LALG-MATRIX-REF-001",
        "author": "Anton, Howard",
        "year": 2010,
        "title": "Elementary Linear Algebra: Applications Version",
        "edition": "10th ed.",
        "pages": "",
        "publisher": "Wiley",
        "link": null
      },
      {
        "id": "MATH-LALG-MATRIX-REF-002",
        "author": "Lay, David C.; Lay, Steven R.; McDonald, Judi J.",
        "year": 2015,
        "title": "Linear Algebra and Its Applications",
        "edition": "5th ed.",
        "pages": "",
        "publisher": "Pearson",
        "link": null
      },
      {
        "id": "MATH-LALG-MATRIX-REF-003",
        "author": "Strang, Gilbert",
        "year": 2016,
        "title": "Introduction to Linear Algebra",
        "edition": "5th ed.",
        "pages": "",
        "publisher": "Wellesley-Cambridge Press",
        "link": null
      },
      {
        "id": "MATH-LALG-MATRIX-REF-004",
        "author": "Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E.",
        "year": 2018,
        "title": "Linear Algebra",
        "edition": "5th ed.",
        "pages": "",
        "publisher": "Pearson",
        "link": null
      },
      {
        "id": "MATH-LALG-MATRIX-REF-005",
        "author": "Axler, Sheldon",
        "year": 2015,
        "title": "Linear Algebra Done Right",
        "edition": "3rd ed.",
        "pages": "",
        "publisher": "Springer",
        "link": null
      },
      {
        "id": "MATH-LALG-MATRIX-REF-006",
        "author": "OpenStax",
        "year": 2021,
        "title": "Algebra and Trigonometry: Matrices and Matrix Operations",
        "edition": null,
        "pages": null,
        "publisher": "OpenStax",
        "link": "https://openstax.org/books/algebra-and-trigonometry/pages/11-5-matrices-and-matrix-operations"
      },
      {
        "id": "MATH-LALG-MATRIX-REF-007",
        "author": "MIT OpenCourseWare",
        "year": 2025,
        "title": "Linear Algebra | Mathematics",
        "edition": null,
        "pages": null,
        "publisher": "MIT OpenCourseWare",
        "link": "https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/"
      }
    ]
  },
  "implementations": {
    "python_numpy": {
      "setup": [
        "import numpy as np"
      ],
      "inputs": [
        "# input matrices",
        "A = np.array([[5, 1, 6], [3, 4, 5], [6, 4, 8]])",
        "B = np.array([[-4, -2, 6], [4, 7, -3], [7, -3, 5]])"
      ],
      "execution": [
        "# calculate",
        "result = np.round(A - B, decimals=5)",
        "result"
      ]
    },
    "python_sympy": {
      "setup": [
        "import sympy as sym"
      ],
      "inputs": [
        "# input matrices",
        "A = sym.Matrix([[5, 1, 6], [3, 4, 5], [6, 4, 8]])",
        "B = sym.Matrix([[-4, -2, 6], [4, 7, -3], [7, -3, 5]])"
      ],
      "execution": [
        "# calculate",
        "result = (A - B).evalf(5)",
        "result"
      ]
    },
    "python_scipy": null,
    "python_pandas": null,
    "python_statsmodels": null,
    "python_scikit_learn": null,
    "python_matplotlib": null,
    "matlab": {
      "setup": [
        "format long g"
      ],
      "inputs": [
        "% input matrices",
        "A = [5 1 6; 3 4 5; 6 4 8];",
        "B = [-4 -2 6; 4 7 -3; 7 -3 5];"
      ],
      "execution": [
        "% calculate",
        "result = round(A - B, 5)"
      ]
    },
    "r": {
      "setup": null,
      "inputs": [
        "# input matrices",
        "A <- matrix(\n\tdata = c(5, 1, 6, 3, 4, 5, 6, 4, 8),\n\tnrow = 3,\n\tbyrow = TRUE\n)",
        "B <- matrix(\n\tdata = c(-4, -2, 6, 4, 7, -3, 7, -3, 5),\n\tnrow = 3,\n\tbyrow = TRUE\n)"
      ],
      "execution": [
        "# calculate",
        "result <- round(A - B, 5)",
        "result"
      ]
    },
    "r_dplyr": null,
    "r_ggplot2": null,
    "stata": null,
    "julia": null,
    "wolfram_language": null,
    "javascript": null
  },
  "exercises": {
    "multiple_choice": [
      {
        "id": "MAT-LALG-SUBTRACT-MC-001",
        "title": "Matrix Addition Definition",
        "question": [
          "What operation is performed when adding two matrices?"
        ],
        "options": [
          {
            "id": "A",
            "text": "Multiply rows by columns",
            "is_correct": false
          },
          {
            "id": "B",
            "text": "Add corresponding entries of the matrices",
            "is_correct": true
          },
          {
            "id": "C",
            "text": "Add the rows together only",
            "is_correct": false
          },
          {
            "id": "D",
            "text": "Multiply corresponding entries",
            "is_correct": false
          }
        ],
        "difficulty": {
          "level": 2,
          "label": "easy",
          "display_name": "Easy",
          "description": "Requires fundamental knowledge and direct application.",
          "age_range": null
        },
        "skill_level": {
          "level": 1,
          "label": "beginner",
          "display_name": "Beginner",
          "description": "Learners with little prior exposure.",
          "age_range": null
        },
        "bloom_level": {
          "level": 1,
          "label": "remember",
          "display_name": "Remember",
          "description": "Recall facts, definitions, and basic information.",
          "age_range": null
        },
        "correct_answer": "Add corresponding entries of the matrices",
        "explanation": [
          "Matrix addition is performed element-by-element.",
          "Each entry is added to the entry in the same position in the other matrix.",
          "The resulting matrix has the same dimensions as the original matrices."
        ],
        "solution_path": null,
        "tags": [
          "linear_algebra",
          "matrices",
          "matrix_addition"
        ]
      },
      {
        "id": "MAT-LALG-SUBTRACT-MC-002",
        "title": "Matrix Addition Requirements",
        "question": [
          "What condition must be satisfied for two matrices to be added?"
        ],
        "options": [
          {
            "id": "A",
            "text": "They must have the same determinant",
            "is_correct": false
          },
          {
            "id": "B",
            "text": "They must have the same trace",
            "is_correct": false
          },
          {
            "id": "C",
            "text": "They must have the same dimensions",
            "is_correct": true
          },
          {
            "id": "D",
            "text": "They must both be square matrices",
            "is_correct": false
          }
        ],
        "difficulty": {
          "level": 2,
          "label": "easy",
          "display_name": "Easy",
          "description": "Requires fundamental knowledge and direct application.",
          "age_range": null
        },
        "skill_level": {
          "level": 1,
          "label": "beginner",
          "display_name": "Beginner",
          "description": "Learners with little prior exposure.",
          "age_range": null
        },
        "bloom_level": {
          "level": 1,
          "label": "remember",
          "display_name": "Remember",
          "description": "Recall facts, definitions, and basic information.",
          "age_range": null
        },
        "correct_answer": "They must have the same dimensions",
        "explanation": [
          "Matrices can only be added when corresponding entries exist.",
          "Both matrices must have the same number of rows and columns.",
          "Different-sized matrices cannot be added."
        ],
        "solution_path": null,
        "tags": [
          "linear_algebra",
          "matrices",
          "dimensions"
        ]
      },
      {
        "id": "MAT-LALG-SUBTRACT-MC-003",
        "title": "Matrix Addition Requirements",
        "question": [
          "Can the following matrices be added?",
          "\\( A = \\left[\\begin{array}{rrr} 5 & 1 & 6 \\\\ 3 & 4 & 5 \\\\ 6 & 4 & 8 \\end{array}\\right] \\quad B = \\left[\\begin{array}{rrr} -4 & -2 \\\\ 4 & 5 \\\\ 7 & -3 \\end{array}\\right] \\)"
        ],
        "options": [
          {
            "id": "A",
            "text": "No",
            "is_correct": true
          },
          {
            "id": "B",
            "text": "Yes",
            "is_correct": false
          }
        ],
        "difficulty": {
          "level": 2,
          "label": "easy",
          "display_name": "Easy",
          "description": "Requires fundamental knowledge and direct application.",
          "age_range": null
        },
        "skill_level": {
          "level": 1,
          "label": "beginner",
          "display_name": "Beginner",
          "description": "Learners with little prior exposure.",
          "age_range": null
        },
        "bloom_level": {
          "level": 1,
          "label": "remember",
          "display_name": "Remember",
          "description": "Recall facts, definitions, and basic information.",
          "age_range": null
        },
        "correct_answer": "No",
        "explanation": [
          "The two matrices have different dimensions. \\( A \\) is a \\( 3 \\times 3 \\) matrix while \\( B \\) is a \\( 3 \\times 2 \\) matrix."
        ],
        "solution_path": null,
        "tags": [
          "linear_algebra",
          "matrices",
          "dimensions"
        ]
      }
    ],
    "practice": [
      {
        "id": "MAT-LALG-SUBTRACT-PR-001",
        "title": "3 x 3 Matrix",
        "question": [
          "Calculate the sum of \\( A + B \\)",
          "\\( A = \\left[\\begin{array}{rrr} 5 & 1 & 6 \\\\ 3 & 4 & 5 \\\\ 6 & 4 & 8 \\end{array}\\right] \\quad B = \\left[\\begin{array}{rrr} -4 & -2 & 6 \\\\ 4 & 7 & -3 \\\\ 7 & -3 & 5 \\end{array}\\right] \\)"
        ],
        "options": null,
        "difficulty": {
          "level": 2,
          "label": "easy",
          "display_name": "Easy",
          "description": "Requires fundamental knowledge and direct application.",
          "age_range": null
        },
        "skill_level": {
          "level": 1,
          "label": "beginner",
          "display_name": "Beginner",
          "description": "Learners with little prior exposure.",
          "age_range": null
        },
        "bloom_level": {
          "level": 3,
          "label": "apply",
          "display_name": "Apply",
          "description": "Use knowledge to solve problems.",
          "age_range": null
        },
        "correct_answer": "\\( {\\color{green}{ans = \\left[\\begin{array}{rrr} 1 & -1 & 12 \\\\ 7 & 11 & 2 \\\\ 13 & 1 & 13 \\end{array}\\right]}} \\)",
        "explanation": null,
        "solution_path": "mathematics/linear-algebra/matrices/arithmetic/subtract?a=TODO&b=TODO",
        "tags": [
          "linear_algebra",
          "matrices",
          "matrix_addition",
          "practice"
        ]
      },
      {
        "id": "MAT-LALG-SUBTRACT-PR-002",
        "title": "Matrix Addition Dimension Check",
        "question": [
          "Can the following matrices be added?",
          "A is a 2x3 matrix.",
          "B is a 3x2 matrix."
        ],
        "options": null,
        "difficulty": {
          "level": 2,
          "label": "easy",
          "display_name": "Easy",
          "description": "Requires fundamental knowledge and direct application.",
          "age_range": null
        },
        "skill_level": {
          "level": 1,
          "label": "beginner",
          "display_name": "Beginner",
          "description": "Learners with little prior exposure.",
          "age_range": null
        },
        "bloom_level": {
          "level": 2,
          "label": "understand",
          "display_name": "Understand",
          "description": "Explain concepts and interpret information.",
          "age_range": null
        },
        "correct_answer": "No, the matrices have different dimensions.",
        "explanation": null,
        "solution_path": "mathematics/linear-algebra/matrices/arithmetic/subtract?a=TODO&b=TODO",
        "tags": [
          "linear_algebra",
          "matrices",
          "matrix_dimensions",
          "practice"
        ]
      }
    ],
    "application": [
      {
        "id": "MAT-LALG-SUBTRACT-AP-001",
        "title": "Inventory Update After Shipments",
        "question": [
          "A store records current inventory levels using a matrix:",
          "I = [[100, 80], [60, 90]]",
          "A new shipment arrives with quantities represented by:",
          "S = [[10, 5], [5, 10]]",
          "Use matrix addition to calculate the updated inventory."
        ],
        "options": null,
        "difficulty": {
          "level": 3,
          "label": "medium",
          "display_name": "Medium",
          "description": "Requires multiple reasoning steps.",
          "age_range": null
        },
        "skill_level": {
          "level": 3,
          "label": "intermediate",
          "display_name": "Intermediate",
          "description": "Learners able to solve standard problems.",
          "age_range": null
        },
        "bloom_level": {
          "level": 3,
          "label": "apply",
          "display_name": "Apply",
          "description": "Use knowledge to solve problems.",
          "age_range": null
        },
        "correct_answer": "[[110, 85], [65, 100]]",
        "explanation": [
          "The shipment matrix and inventory matrix have the same dimensions.",
          "Add corresponding entries: I + S.",
          "[[100+10, 80+5], [60+5, 90+10]] = [[110, 85], [65, 100]]."
        ],
        "solution_path": "mathematics/linear-algebra/matrices/arithmetic/subtract?a=TODO&b=TODO",
        "tags": [
          "linear_algebra",
          "matrices",
          "inventory",
          "matrix_addition",
          "application"
        ]
      },
      {
        "id": "MAT-LALG-SUBTRACT-AP-002",
        "title": "Combining Regional Sales Data",
        "question": [
          "A company stores monthly sales data from two regions as matrices.",
          "Region A sales:",
          "A = [[120, 150], [200, 250]]",
          "Region B sales:",
          "B = [[100, 130], [180, 220]]",
          "Combine the two regional sales records using matrix addition."
        ],
        "options": null,
        "difficulty": {
          "level": 3,
          "label": "medium",
          "display_name": "Medium",
          "description": "Requires multiple reasoning steps.",
          "age_range": null
        },
        "skill_level": {
          "level": 3,
          "label": "intermediate",
          "display_name": "Intermediate",
          "description": "Learners able to solve standard problems.",
          "age_range": null
        },
        "bloom_level": {
          "level": 3,
          "label": "apply",
          "display_name": "Apply",
          "description": "Use knowledge to solve problems.",
          "age_range": null
        },
        "correct_answer": "[[220, 280], [380, 470]]",
        "explanation": [
          "Both matrices represent the same categories and months.",
          "Corresponding entries are added together.",
          "A + B = [[120+100, 150+130], [200+180, 250+220]]."
        ],
        "solution_path": "mathematics/linear-algebra/matrices/arithmetic/subtract?a=TODO&b=TODO",
        "tags": [
          "linear_algebra",
          "matrices",
          "economics",
          "sales",
          "matrix_addition",
          "application"
        ]
      }
    ]
  },
  "additional": {
    "remarks": [],
    "warnings": [],
    "notices": []
  }
}