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# Show that the matrix $A + B$ is symmetric or skew symmetric according as $A$and $B$ are symmetric or skew symmetric. Verified
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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.
Last updated date: 20th Sep 2023
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