Question

If $A$ is a skew symmetric matrix then ${A^T}$
(A) $- A$
(B) $A$
(C)$0$
(D)diagonal matrix

Answer 1

A skew symmetric matrix is a square matrix whose transpose is its negation, i.e., it satisfies the condition: $- A = {A^T}$, where ${A^T}$ is the transpose of matrix $A$.

Complete step-by-step answer:
Here it is necessary to understand the transpose of a matrix and skew a symmetric matrix to find the solution of a given question.
Step 1: Transpose of a matrix- Let $A$ be a matrix of order $m \times n$. Then, the $n \times m$ order matrix obtained by interchanging the rows and columns of $A$ is called the transpose of $A$ and is denoted by ${A^T}$.
For ex- If A = {\left[ {\begin{array}{*{20}{c}} 1&3&5 \\\ 2&4&6 \end{array}} \right]_{2 \times 3}}, then {A^T} = {\left[ {\begin{array}{*{20}{c}} 1&2 \\\ 3&4 \\\ 5&6 \end{array}} \right]_{3 \times 2}}
Step2: Skew symmetric matrix- A matrix can be skew symmetric only if it is square. If the transpose of a matrix is equal to the negative of itself, the matrix is said to be skew symmetric. This means that for a matrix to be skew symmetric,
${A^T} = - A$
For ex- If A = \left[ {\begin{array}{*{20}{c}} 0&2&3 \\\ { - 2}&0&4 \\\ { - 3}&{ - 4}&0 \end{array}} \right], then {A^T} = \left[ {\begin{array}{*{20}{c}} 0&{ - 2}&{ - 3} \\\ 2&0&{ - 4} \\\ 3&4&0 \end{array}} \right] $= - A$
Hence, for a skew symmetric matrix: ${A^T} = - A$
So, option (A) is the correct answer.

Note: It may be noted that the diagonal elements of a skew symmetric matrix are always equal to zero and therefore the sum of elements of the main diagonals is also equal to zero.