CBSE Sample Paper 2025-26 · Section A · 1 mark

Q3: Value of $\frac{q + t}{p + r}$ for a skew-symmetric matrix

Sample 2025-26Ch 3 · Matrices

The question

If the matrix $A = \begin{bmatrix} 0 & r & -2 \\ 3 & p & t \\ q & -4 & 0 \end{bmatrix}$ is skew-symmetric, then value of $\frac{q + t}{p + r}$ is….

(A) $-2$

(B) $0$

(D) $2$

Exam answer

$A$ is skew-symmetric, so $a_{ji} = -a_{ij}$ and $a_{ii} = 0$ ⁠.

Why?

Skew-symmetric means $A' = -A$ ⁠. Comparing the entries in position $(j, i)$ gives $a_{ji} = -a_{ij}$ ⁠.

On the diagonal $i = j$ ⁠, so $a_{ii} = -a_{ii}$ ⁠, which forces $a_{ii} = 0$ ⁠.

Comparing entries,

\begin{aligned} p &= a_{22} = 0 \\ r &= a_{12} = -a_{21} = -3 \\ q &= a_{31} = -a_{13} = 2 \\ t &= a_{23} = -a_{32} = 4 \end{aligned}

Why?

$p$ sits on the diagonal (row 2, column 2), so it is 0.

Read the partners from the matrix: $a_{21} = 3$ (row 2, column 1), $a_{13} = -2$ and $a_{32} = -4$ ⁠.

Watch out: $q = -(-2) = 2$ and $t = -(-4) = 4$ ⁠. Two minus signs make a plus.

So $p = 0$ ⁠, $r = -3$ ⁠, $q = 2$ ⁠, $t = 4$ ⁠.

\frac{q + t}{p + r} = \frac{2 + 4}{0 - 3} = -2

Why?

The numerator is $2 + 4 = 6$ and the denominator is $0 + (-3) = -3$ ⁠, so the value is $\frac{6}{-3} = -2$ ⁠.

Answer: (A) $-2$

Where this comes from

Chapter 3, Matrices: Skew-symmetric matrices.

Question type: Transpose

How to spot it: click to see

Type 8. Transpose: symmetric or skew symmetric?

Spot it: "show that $A$ is symmetric", "show that $A$ is skew symmetric", or "find $A'$ ".

Here: $A$ is skew symmetric, so $p$ on the diagonal is 0, and each other entry is minus its mirror: $r = -3$ ⁠, $q = 2$ ⁠, $t = 4$ ⁠. Then $\frac{q + t}{p + r} = -2$ ⁠.

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