Sample 2024-25 Class 12 Maths Q7 Solution: a + b + c for a skew-symmetric matrix

The question

If

A = \begin{bmatrix} 0 & 1 & c \\ -1 & a & -b \\ 2 & 3 & 0 \end{bmatrix}

is a skew-symmetric matrix then the value of $a + b + c =$

(A) $1$

(B) $2$

(C) $3$

(D) $4$

Exam answer

$A$ is skew-symmetric, so $A^T = -A$ ⁠, that is $a_{ij} = -a_{ji}$ ⁠.

Why?

Transposing swaps $a_{ij}$ and $a_{ji}$ ⁠. So $A^T = -A$ says that each entry is minus its mirror image across the main diagonal.

On the diagonal an entry is its own mirror image: $a_{ii} = -a_{ii}$ gives $2a_{ii} = 0$ ⁠, so every diagonal entry is 0.

Here $a_{22} = a$ ⁠, $a_{13} = c$ ⁠, $a_{31} = 2$ ⁠, $a_{23} = -b$ and $a_{32} = 3$ ⁠.

\begin{aligned} a_{22} = -a_{22} &\Rightarrow a = 0 \\ a_{13} = -a_{31} &\Rightarrow c = -2 \\ a_{23} = -a_{32} &\Rightarrow -b = -3 \\ &\Rightarrow b = 3 \end{aligned}

Why?

$a_{ij}$ is the entry in row $i$ ⁠, column $j$ ⁠. So $c$ (row 1, column 3) pairs with 2 (row 3, column 1), and $-b$ (row 2, column 3) pairs with 3 (row 3, column 2).

\begin{aligned} a = -a &\Rightarrow 2a = 0 \Rightarrow a = 0 \\ c = -(2) &\Rightarrow c = -2 \\ -b = -(3) &\Rightarrow -b = -3 \\ &\Rightarrow b = 3 \quad (\times\,(-1)) \end{aligned}

Watch out for the sign of $b$ ⁠: the entry is $-b$ ⁠, so $-b = -3$ gives $b = 3$ ⁠, not $-3$ ⁠.

The remaining pair already fits: $a_{12} = 1$ and $a_{21} = -1$ ⁠.

So $a + b + c = 0 + 3 - 2 = 1$ ⁠.

Answer: (A) $1$

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 match each entry with minus its mirror: $a$ is on the diagonal, so $a = 0$ ⁠; $c = -2$ and $-b = -3$ ⁠. So $a + b + c = 1$ ⁠.

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