Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew-symmetric,then the matrix $AB - BA$ is

If $A$ is a skew-symmetric matrix,then (given $n \in N$):
$1$. $A^{2n}$ is a skew-symmetric matrix.
$2$. $A^{2n+1}$ is a skew-symmetric matrix.

Find $\frac{1}{2}(A+A^{\prime})$ and $\frac{1}{2}(A-A^{\prime}),$ when $A=\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right].$

If $ A=\begin{bmatrix} \cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{bmatrix} $ and $ A+A^{T}=I $,where $ I $ is the $ 2 \times 2 $ identity matrix and $ A^{T} $ is the transpose of $ A $,then the value of $ \theta $ is equal to

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix: $\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

If $P$ and $Q$ are symmetric matrices of the same order,then $PQ - QP$ is