If $A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$,then $A A^T$ is a

If $A^{\prime}=\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$,then verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.

If $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$,then $A' = $ . . . . . . .

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 $A^{\prime}=\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$,then verify that $(A-B)^{\prime}=A^{\prime}-B^{\prime}$.

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