Symmetric and Skew Symmetric Matrices

If the matrix $A$ is both symmetric and skew symmetric, then

Show that the matrix $B^{\text{'}}AB$ is symmetric or skew-symmetric according to $A$ is symmetric or skew-symmetric.

If $A$ and $B$ are symmetric matrices, prove that $AB-BA$ is a skew symmetric matrix.

If $A,B$ are symmetric matrices of the same order, then $AB-BA$ is a

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix

$\left[\begin{array}{ll}1& 5\\ -1& 2\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix:

$\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix:

$\left[\begin{array}{ccc}6& -2& 2\\ -2& 3& -1\\ 2& -1& 3\end{array}\right]$

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix:

$\left[\begin{array}{cc}3& 5\\ 1& -1\end{array}\right]$

For the matrix $A=\left[\begin{array}{ll}1& 5\\ 6& 7\end{array}\right]$, verify that $\left(A-A^{\text{'}}\right)$ is a skew symmetric matrix

For the matrix $A=\left[\begin{array}{ll}1& 5\\ 6& 7\end{array}\right]$, verify that $\left(A+A^{\text{'}}\right)$ is a symmetric matrix

Show that the matrix $A=\left[\begin{array}{ccc}0& 1& -1\\ -1& 0& 1\\ 1& -1& 0\end{array}\right]$ is a skew-symmetric matrix.

Show that the matrix $A=\left[\begin{array}{ccc}1& -1& 5\\ -1& 2& 1\\ 5& 1& 3\end{array}\right]$ is a symmetric matrix.