Question

Show that the matrix $A+B$ is symmetric or skew symmetric according as $A$and $B$ are symmetric or skew symmetric.

Answer 1

Hint: - Use the properties of matrix transpose and addition of matrix.

Since, ${\left(A+B\right)}^{\prime }=A^{\prime }+B^{\prime }$
For any symmetric matrix $M$ we know that $M^{\prime }=M$.
If both $A$ and $B$ are symmetric.
$\Rightarrow A^{\prime }=A\mathrm{\&}{B}^{\prime }=B$
For $A+B$ matrix, we have
$$\begin{gathered} \Rightarrow {\left(A+B\right)}^{\prime }=A^{\prime }+B^{\prime } \\ \Rightarrow A^{\prime }+B^{\prime }=A+B\left[\because A^{\prime }=A\mathrm{\&}{B}^{\prime }=B\right] \end{gathered}$$
$\therefore$$A+B$ is symmetric, as${\left(A+B\right)}^{\prime }=A+B$
For any skew symmetric matrix $M$ we know that $M^{\prime }=-M$ .
If both $A$ and $B$ are skew symmetric.
$\Rightarrow A^{\prime }=-A\mathrm{\&}{B}^{\prime }=-B$
For $A+B$ matrix, we have
$$\begin{gathered} \Rightarrow {\left(A+B\right)}^{\prime }=A^{\prime }+B^{\prime } \\ \Rightarrow A^{\prime }+B^{\prime }=-A-B\left[\because A^{\prime }=A\mathrm{\&}{B}^{\prime }=B\right] \\ \Rightarrow -\left(A+B\right) \end{gathered}$$
$\therefore$$A+B$ is skew symmetric, as${\left(A+B\right)}^{\prime }=-\left(A+B\right)$

Note: Symmetric matrix is a square matrix that is equal to its transpose. Only a square matrix can be symmetric whereas a matrix is called skew symmetric if and only if it is opposite of its transpose.