Sample 2022-23 Class 12 Maths Q1 Solution: Diagonal elements of a skew-symmetric matrix

The question

If $A = [a_{ij}]$ is a skew-symmetric matrix of order n, then

(a) $a_{ij} = \frac{1}{a_{ji}}\ \forall\ i, j$

(b) $a_{ij} \ne 0\ \forall\ i, j$

(c) $a_{ij} = 0$ ⁠, where $i = j$

(d) $a_{ij} \ne 0$ where $i = j$

Exam answer

A is skew-symmetric, so $A' = -A$ ⁠, that is $a_{ji} = -a_{ij}$ for all $i, j$ ⁠.

Why?

The transpose $A'$ swaps rows and columns, so the $(i, j)$ entry of $A'$ is $a_{ji}$ ⁠. Writing $A' = -A$ entry by entry gives

a_{ji} = -a_{ij}

Compare a symmetric matrix, where $A' = A$ gives $a_{ji} = a_{ij}$ ⁠.

For $i = j$ ⁠: $a_{ii} = -a_{ii}$ ⁠, so $2a_{ii} = 0$ and $a_{ii} = 0$ ⁠.

Why?

Putting $j = i$ picks out a diagonal element. The only number equal to its own negative is 0, so every diagonal element is 0.

So (c) is true, and (b) and (d) are false. (a) is false too: with $a_{12} = 2$ ⁠, $a_{21} = -2$ ⁠, we get $\frac{1}{a_{21}} = -\frac{1}{2} \ne a_{12}$ ⁠.

Why?

Watch out: in a skew-symmetric matrix the mirror entry is the negative, $a_{ij} = -a_{ji}$ ⁠, not the reciprocal. For example

\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}

is skew-symmetric, with $a_{12} = 2$ and $a_{21} = -2$ ⁠.

Answer: (c) $a_{ij} = 0$ ⁠, where $i = j$

Where this comes from

Chapter 3, Matrices: Skew-symmetric matrix: diagonal elements.

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 skew symmetric: $A' = -A$ gives $a_{ii} = -a_{ii}$ on the diagonal, so every diagonal entry is 0.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Q1: Diagonal elements of a skew-symmetric matrix