question_answer

Show that a matrix which is both symmetric as-well-as skew-symmetric is a null matrix.

Answer:

Let A is a matrix which is both symmetric as-well-as skew-symmetric matrix. \[\therefore \]                  \[A'=A\]                       ?(i) And                  \[A'=-\,A\]                   ?(ii) On comparing Eqs. (i) and (ii), we get \[A=-\,A\]             \[\Rightarrow \]   \[A+A=0\Rightarrow 2A=0\Rightarrow A=0\] Hence, matrix which is both symmetric as well as skew-symmetric is a null matrix.