Question

If the matrix $A$ is both symmetric and skew-symmetric, then
A) $A$ is a diagonal matrix
B) $A$ is a zero matrix
C) $A$ is a square matrix
D) None of these

Answer 1

Here, we will first use that when a matrix $A$ is symmetric, then ${A^T} = A$ and if a matrix $A$ is skew-symmetric, then ${A^T} = - A$ and the diagonal elements are also zero. Then we will find a matrix, which fits it all.

Complete step by step solution: We are given that the matrix $A$ is both symmetric and skew-symmetric.

We know that if a matrix $A$ is symmetric, then ${A^T} = A$ and if a matrix $A$ is skew symmetric, then ${A^T} = - A$ and the diagonal elements are also zero.

Since we are given that a matrix $A$ is both symmetric and skew-symmetric, then we have

$\Rightarrow A = {A^T} = - A$

But the above expression is only possible if $A$ is a zero matrix.

If A = \left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right], find the value of ${A^T}$ and $- A$.

\Rightarrow {A^T} = {\left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right]^T} \\\ \Rightarrow {A^T} = \left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right] \\\ \Rightarrow - A = - \left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right] \\\ \Rightarrow - A = \left[ {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right] \\\

Thus, the zero matrices are the only matrix, which is both symmetric and skew-symmetric matrix.

Hence, option B is correct.

Note: While solving these types of problems, students should know that the symmetric matrix is a square matrix that is equal to its transpose and the skew-symmetric matrix is a square matrix that is equal to the negative of its transpose. One should know a transpose of a matrix is an operator which flips a matrix over its diagonal, where it switches the row and column indices of the matrix $A$ by producing another matrix, denoted by ${A^T}$.