If $A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 5 \\ 2 & -5 & 0 \end{bmatrix}$,then

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

The matrix $A = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}$ is

If $A$ is a symmetric matrix,then the matrix $M'AM$ is

Let $2A+B = \begin{bmatrix} 1 & 0 & 3 \\ -1 & 4 & 6 \\ 2 & 5 & 2 \end{bmatrix}$ and $A-2B = \begin{bmatrix} 2 & -1 & 5 \\ 0 & 3 & 6 \\ 1 & 2 & 1 \end{bmatrix}$. Then $Tr(A) - Tr(B)$ has the value equal to (where $Tr(A)$ denotes the trace of matrix $A$).

If $A$ and $B$ are symmetric matrices of the same order such that $AB+BA=X$ and $AB-BA=Y$,then $(XY)^{T}=$