Question

In a skew-symmetric matrix, the diagonal elements are all.

• A. One.
• B. Zero.
• C. Different from each other.
• D. Non-zero.

Answer 1

The correct option is B

Zero.

Explanation for the correct option:

Step $1:$ Concept to be used.

Properties of the skew-symmetric matrix:

1. A skew-symmetric matrix is always a square matrix.
2. The transpose of the skew-symmetric matrix is the negation of itself.

Step $2:$ Prove that all the diagonal elements of the skew-symmetric matrix are zero.

Since the transpose of the skew-symmetric matrix is the negation of itself and also skew-symmetric matrix is always a square matrix.

Assume that, $A$ be a skew-symmetric matrix.

So, $A^T=-A$

$\Rightarrow A_{ji}=-A_{ij}$

Since for diagonal elements $i=j$.

So,

$$\begin{gathered} A_{ii}=-A_{ii} \\ \Rightarrow A_{ii}+A_{ii}=0 \\ \Rightarrow 2A_{ii}=0 \\ \Rightarrow A_{ii}=0 \end{gathered}$$

Therefore, all the diagonal elements of a skew-symmetric matrix are always zero.

Hence, option $B$ is the correct answer.