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Matrix Operations

Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions.

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Student-friendly explanation

Matrices of the same order can be added or subtracted by combining corresponding entries. A scalar multiplies every entry of a matrix. For product AB, if A has order m x n and B has order n x p, then AB exists and has order m x p. Each entry is found by row-column multiplication.

How to write this in exams

  1. 1

    Start with the exact idea

    Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions.

  2. 2

    Then show how to use it

    Write the order of each matrix. For addition or subtraction, confirm same order and combine corresponding entries. For scalar multiplication, multiply every entry. For product, confirm columns of first equal rows of second, then compute each entry by row-column dot product.

  3. 3

    Add one concrete example

    If A = [[1, 2], [3, 4]] and B = [[5, 0], [-1, 2]], then A + B = [[6, 2], [2, 6]] and 2A = [[2, 4], [6, 8]].

  4. 4

    Avoid this incomplete answer

    Forgetting to multiply the scalar with every entry, especially negative or zero entries.

Definition

Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions.

Example

If A = [[1, 2], [3, 4]] and B = [[5, 0], [-1, 2]], then A + B = [[6, 2], [2, 6]] and 2A = [[2, 4], [6, 8]].

Rule to remember

Addition condition: A and B must have the same order. Scalar multiplication: (kA)_ij = k a_ij. Product condition: if A is m x n and B is n x p, then AB is m x p with (AB)_ij = sum over r of a_ir b_rj.

Memory hook

For AB, the row belongs to A and the column belongs to B.

Examples and method

Worked example

Let A = [[1, 2], [0, -1]] and B = [[3, 1], [4, 2]]. AB = [[1 x 3 + 2 x 4, 1 x 1 + 2 x 2], [0 x 3 + (-1) x 4, 0 x 1 + (-1) x 2]] = [[11, 5], [-4, -2]].

Method to apply

Write the order of each matrix. For addition or subtraction, confirm same order and combine corresponding entries. For scalar multiplication, multiply every entry. For product, confirm columns of first equal rows of second, then compute each entry by row-column dot product.

Diagram support

No standalone diagram is required, but a row-column marking in multiplication can help students align entries.

How CBSE asks it

Questions ask students to compute A+B, kA, AB, or form equations by comparing entries after operations.

Avoid common mistakes

Common confusion

Students often multiply corresponding entries and call it matrix multiplication. Correct multiplication uses row of the first matrix with column of the second matrix.

Common wrong answer

Forgetting to multiply the scalar with every entry, especially negative or zero entries.

Exam tip

In multiplication, check the inner dimensions first. If they match, the outer dimensions give the order of the product.

Quick check

If A is of order 2 x 3 and B is of order 3 x 4, what is the order of AB?

AB exists because the inner dimensions 3 and 3 match. Its order is 2 x 4.

Answer writing and exam use

1-mark answer

Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions.

2-mark answer

Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions. Addition condition: A and B must have the same order. Scalar multiplication: (kA)_ij = k a_ij. Product condition: if A is m x n and B is n x p, then AB is m x p with (AB)_ij = sum over r of a_ir b_rj. If A = [[1, 2], [3, 4]] and B = [[5, 0], [-1, 2]], then A + B = [[6, 2], [2, 6]] and 2A = [[2, 4], [6, 8]].

3-mark answer

Matrices of the same order can be added or subtracted by combining corresponding entries. A scalar multiplies every entry of a matrix. For product AB, if A has order m x n and B has order n x p, then AB exists and has order m x p. Each entry is found by row-column multiplication. Addition condition: A and B must have the same order. Scalar multiplication: (kA)_ij = k a_ij. Product condition: if A is m x n and B is n x p, then AB is m x p with (AB)_ij = sum over r of a_ir b_rj. Let A = [[1, 2], [0, -1]] and B = [[3, 1], [4, 2]]. AB = [[1 x 3 + 2 x 4, 1 x 1 + 2 x 2], [0 x 3 + (-1) x 4, 0 x 1 + (-1) x 2]] = [[11, 5], [-4, -2]]. Questions ask students to compute A+B, kA, AB, or form equations by comparing entries after operations. Forgetting to multiply the scalar with every entry, especially negative or zero entries.
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