medium difficulty7 min

Matrix Operations

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

In one line: For AB, the row belongs to A and the column belongs to B.

Practice This Concept

Understand it

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.

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]].

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

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

Method you can reuse

1Write the order of each matrix.
2For addition or subtraction, confirm same order and combine corresponding entries.
3For scalar multiplication, multiply every entry.
4For product, confirm columns of first equal rows of second, then compute each entry by row-column dot product.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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

1
Enough for 1 mark
Matrix operations include addition, subtraction, scalar multiplication, and multiplication of matrices, each governed by order conditions.
2
Add for 2 marks
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]].
3
Add for 3 marks
Matrices of the same order can be added or subtracted by combining corresponding entries.

MCQ Quiz

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