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

Hint: 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 \[{A^T}\].