Assume $X, Y, Z, W$ and $P$ are the matrices of order $2 \times n, 3 \times k, 2 \times p, n \times 3$ and $p \times k$ respectively. The restriction on $n, k$ and $p$ so that $PY + WY$ will be defined are:

Let $A=\begin{bmatrix} a & 3 & 5 \\ 5 & -1 & 3 \\ 2 & 3 & -4 \end{bmatrix}$ and $B=\begin{bmatrix} b & 1 & 4 \\ 4 & c & 1 \\ -3 & 1 & d \end{bmatrix}$. If the trace of $A$ is $-4$ and $AB=\begin{bmatrix} -1 & 0 & 17 \\ -3 & 10 & 25 \\ 28 & -8 & 3 \end{bmatrix}$, then $a+b+c+d=$

If $A$ and $B$ are two matrices and $(A + B)(A - B) = A^2 - B^2$,then

Let $A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix}$,$B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix}$,and $C = \begin{bmatrix} 3 & 1 & 2 \end{bmatrix}$. The expression which is not defined is:

If $f(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & -\cos \theta \end{bmatrix}$,then $f\left(\frac{\pi}{6}\right) = $ . . . . . . .

The order of matrix $A$ is $m \times n$ and for matrix $B$,if $AB^{\prime}$ and $B^{\prime}A$ are defined,then the order of matrix $B$ is . . . . . . .