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CBSE Board Examination 2025 · Section A · 1 mark

Q9: Which of AB, AC, BA is defined?

Board 2025Ch 3 · Matrices
The question

Let A=[1−2−104−1−321]A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}⁠, B=[−2−5−7]B = \begin{bmatrix} -2 \\ -5 \\ -7 \end{bmatrix}⁠, C=[987]C = \begin{bmatrix} 9 & 8 & 7 \end{bmatrix}⁠, which of the following is defined ?

(A) Only AB

(B) Only AC

(C) Only BA

(D) All AB, AC and BA

Exam answer

Orders: A is 3×33 \times 3⁠, B is 3×13 \times 1⁠, C is 1×31 \times 3⁠.

Why?

The order m×nm \times n means m rows and n columns. B is a single column of 3 entries, so it is 3×13 \times 1⁠. C is a single row of 3 entries, so it is 1×31 \times 3⁠.

PQ is defined only when the number of columns of P equals the number of rows of Q.

Why?

Each entry of PQ pairs a row of P with a column of Q, entry by entry, so the two must have the same length. In terms of orders:

(m×n)(n×p)=m×p(m \times n)(n \times p) = m \times p

The two inside numbers must match; the outside numbers give the order of the product.

AB is defined: A has 3 columns and B has 3 rows (AB is 3×13 \times 1⁠).

AC is not defined: A has 3 columns, C has 1 row.

BA is not defined: B has 1 column, A has 3 rows.

So only AB is defined, and (D) is false.

Why?

Watch out: the order of multiplication matters. AB is defined but BA is not. Similarly, CA would be defined, (1×3)(3×3)(1 \times 3)(3 \times 3)⁠, even though AC is not.

Answer: (A) Only AB

Where this comes from

Chapter 3, Matrices: When a matrix product is defined.

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: AA is 3×33 \times 3⁠, BB is 3×13 \times 1⁠, CC is 1×31 \times 3⁠. Only in ABAB do the inside numbers match (3 and 3).

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