If $X_{4 \times 3}$,$Y_{4 \times 3}$ and $P_{2 \times 3}$ are matrices,then the order of the matrix $\left[P(X^T Y)^{-1} P^T\right]^T$ is

If $A = \begin{bmatrix} \alpha - 1 \\ 0 \\ 0 \end{bmatrix}$ and $B = \begin{bmatrix} \alpha + 1 \\ 0 \\ 0 \end{bmatrix}$ are two matrices,then $AB^T$ is a non-zero matrix for $|\alpha|$ not equal to:

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

If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime}B$ and $BA^{\prime}$ are both defined,then $B$ is of the type:

If $M = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ and $M^2 - \lambda M - I_2 = 0$,then $\lambda = $

Find $X$ and $Y,$ if $2X+3Y=\left[\begin{array}{cc}2 & 3 \\ 4 & 0\end{array}\right]$ and $3X+2Y=\left[\begin{array}{cc}2 & -2 \\ -1 & 5\end{array}\right].$