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