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

Hint: A square matrix $$A=\left[a_{ij}\right]$$is said to be skew symmetric matrix if
$A^{\prime }=-A$ or $A=-A^{\prime }$, that is $$a_{ij}=-a_{ji}$$for all possible values of $i$ and $j$.
In transpose of a matrix, columns and rows are interchanged. Transpose denoted by: $A^{\prime }\text{ (or }{A}^T)$. For example:
If $$A={\left[\begin{array}{c}3\\ \sqrt{3}\\ 0\end{array}\begin{array}{c}5\\ 1\\ {\displaystyle \frac{-1}{5}}\end{array}\right]}_{3\times 2}$$
Then $$A^{\prime }={\left[\begin{array}{c}3\\ 5\end{array}\begin{array}{c}\sqrt{3}\\ 1\end{array}\begin{array}{c}0\\ {\displaystyle \frac{-1}{5}}\end{array}\right]}_{2\times 3}$$

Complete step-by-step answer:
Step 1: Consider a square matrix $$A=\left[a_{ij}\right]$$
Where $i$: row number and $j$: column number.
Step 2: Condition for skew symmetric matrix:
$A^{\prime }=-A$
Here,$A^{\prime }$is transpose of matrix A
i.e. $$a_{ij}=-a_{ji}$$
Step 3: Now, if we put $i=j$,
We have, $$a_{ii}=-a_{ii}$$
$$\begin{gathered} \therefore 2a_{ii}=0 \\ \Rightarrow a_{ii}=0 \end{gathered}$$ for all $i^{\prime }s$.
Step 4: diagonal elements of a square matrix
In the square matrix $$A=\left[a_{ij}\right]$$
$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$
Elements $a_{11},a_{22},a_{33}$ are diagonal elements.
$a_{ii}=0$
$\Rightarrow a_{11}=a_{22}=a_{33}=0$

All the diagonal elements of the skew symmetric matrix are zero. Thus, the correct option is (B).

Note: Another way to understand the solution.
We have a theorem: Any square matrix A with real number entries, $A-A^{\prime }$is a skew symmetric matrix.
Example question: The skew symmetric matrix of matrix $B=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$.