Sample 2025-26 Class 12 Maths Q2 Solution: Orders m, n, p, q when AB and AC are square

The question

If for three matrices $A = [a_{ij}]_{m \times 4}$ ⁠, $B = [b_{ij}]_{n \times 3}$ and $C = [c_{ij}]_{p \times q}$ products $AB$ and $AC$ both are defined and are square matrices of same order, then value of $m, n, p$ and $q$ are:

(A) $m = q = 3$ and $n = p = 4$

(B) $m = 2, q = 3$ and $n = p = 4$

(C) $m = q = 4$ and $n = p = 3$

(D) $m = 4, p = 2$ and $n = q = 3$

Exam answer

$AB$ is defined, so $n = 4$ ⁠, and $AB$ is of order $m \times 3$ ⁠.

Why?

A product $PQ$ is defined only when the number of columns of $P$ equals the number of rows of $Q$ ⁠. $A$ has 4 columns and $B$ has $n$ rows, so $n = 4$ ⁠.

The product takes its rows from the first matrix and its columns from the second:

(m \times 4)(4 \times 3) \to m \times 3

$AC$ is defined, so $p = 4$ ⁠, and $AC$ is of order $m \times q$ ⁠.

Why?

The same rule: $A$ has 4 columns and $C$ has $p$ rows, so $p = 4$ ⁠.

(m \times 4)(4 \times q) \to m \times q

$AB$ is square, so $m = 3$ ⁠. $AC$ is of the same order, so $m \times q = 3 \times 3$ and $q = 3$ ⁠.

Why?

A square matrix has as many rows as columns. $AB$ has 3 columns, so it needs 3 rows: $m = 3$ ⁠.

Same order means the same number of rows and the same number of columns. So $AC$ is also $3 \times 3$ ⁠, which gives $q = 3$ ⁠.

So $m = q = 3$ and $n = p = 4$ ⁠. Option (B) has $m = 2$ ⁠, and options (C) and (D) have $n = 3$ ⁠, so they fail.

Answer: (A) $m = q = 3$ and $n = p = 4$

Where this comes from

Chapter 3, Matrices: Order of a product of matrices.

Question type: One-mark MCQs

How to spot it: click to see

Type 6. One-mark MCQs: orders and definitions

Spot it: four options, asking when a sum or product is defined, what order it has, or what a word such as symmetric or inverse means.

Here: Write the orders: $AB$ is $(m \times 4)(n \times 3)$ ⁠, so $n = 4$ ⁠, and $AC$ needs $p = 4$ ⁠. $AB$ is $m \times 3$ and square, so $m = 3$ ⁠; $AC$ is $m \times q$ of the same order, so $q = 3$ ⁠.

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Q2: Orders m,n,p,qm, n, p, qm,n,p,q when ABABAB and ACACAC are square

Q2: Orders $m, n, p, q$ when $AB$ and $AC$ are square