Question

If a matrix A is both symmetric and skew symmetric, then
a) A is diagonal matrix
b) A is a zero matrix
c) A is scalar matrix
d) A is square matrix

Answer 1

A matrix ‘A’ is symmetric if and only if $A={{A}^{T}}$ and a matrix ‘A’ is skew symmetric if and only if $A=-A$. In order to answer this question, we will try to find a matrix that fits the condition of $A={{A}^{T}}=-A$. You should remember that all the diagonal elements of skew symmetric are always equal to zero.

Complete step by step solution:
We have given that the matrix A is symmetric and skew symmetric both.
If matrix A is symmetric,
Therefore,
$\Rightarrow {{A}^{T}}=A$------ (1)
If matrix A is skew symmetric,
Therefore,
$\Rightarrow {{A}^{T}}=-A$------ (2)
And in skew symmetric, diagonal elements should always be zero.
On, comparing equation (1) and equation (2), we get
$\Rightarrow A={{A}^{T}}=-A$
It is also given that the matrix A is both symmetric and skew symmetric both.
The above condition is only possible, when A is a zero matrix i.e.

0 & 0 \\\ 0 & 0 \\\ \end{matrix} \right)$$ Now, finding the $${{A}^{T}}$$matrix $$\Rightarrow {{A}^{T}}={{\left( \begin{matrix} 0 & 0 \\\ 0 & 0 \\\ \end{matrix} \right)}^{T}}$$ $$\Rightarrow {{A}^{T}}=\left( \begin{matrix} 0 & 0 \\\ 0 & 0 \\\ \end{matrix} \right)$$ (as $${{0}^{any\ number}}=0$$) Now, finding the –A MATRIX $$\Rightarrow -A=-\left( \begin{matrix} 0 & 0 \\\ 0 & 0 \\\ \end{matrix} \right)$$ $$\Rightarrow -A=\left( \begin{matrix} 0 & 0 \\\ 0 & 0 \\\ \end{matrix} \right)$$ (as zero is neither negative nor positive) Therefore, if A is a zero matrix then$${{A}^{T}}=A=-A$$. **Hence the option (b) is the correct answer.** **Note:** While solving these types of questions, should always remember what symmetric matrix is and what skew symmetric matrix is. A symmetric matrix is a square matrix which is equal to the transpose. One should always remember that the transpose of a matrix refers to the flipping of the matrix over the diagonal.