CBSE Board Examination 2025 · Section A · 1 mark

Q13: $y^x$ when a matrix is scalar

Board 2025Ch 3 · Matrices

The question

If $A = \begin{bmatrix} 7 & 0 & x \\ 0 & 7 & 0 \\ 0 & 0 & y \end{bmatrix}$ is a scalar matrix, then $y^x$ is equal to

(A) $0$

(B) $1$

(D) $\pm 7$

Exam answer

In a scalar matrix every non-diagonal entry is 0 and the diagonal entries are equal.

Why?

A scalar matrix is a diagonal matrix with all its diagonal entries equal, that is, kI for some number k. Here it would be 7I.

The entry $a_{13} = x$ is not on the diagonal, so it must be 0.

The diagonal entries 7, 7 and y must be equal.

So $x = 0$ and $y = 7$ ⁠.

Why?

$a_{13}$ means row 1, column 3. The diagonal entries are $a_{11} = 7$ ⁠, $a_{22} = 7$ and $a_{33} = y$ ⁠.

$\therefore\ y^x = 7^0 = 1$

Why?

Any non-zero number to the power 0 is 1.

Watch out: the question asks for $y^x$ ⁠, not y. The value 7 is option (C), a trap.

Answer: (B) $1$

Where this comes from

Chapter 3, Matrices: Scalar matrix.

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: The word here is scalar: $x$ sits off the diagonal, so $x = 0$ ⁠, and the diagonal $7, 7, y$ must be equal, so $y = 7$ ⁠. Then $y^x = 7^0 = 1$ ⁠.

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Q13: $y^x$ when a matrix is scalar