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

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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.

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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