The number of cubic polynomials $P(x)$ satisfying $P(1)=2, P(2)=4, P(3)=6, P(4)=8$ is

Let $A$, $B$ and $C$ be three $2 \times 2$ matrices with real entries such that $B = (I + A)^{-1}$ and $A + C = I$. If $BC = \begin{bmatrix} 1 & -5 \\ -1 & 2 \end{bmatrix}$ and $CB \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 12 \\ -6 \end{bmatrix}$, then $x_1 + x_2$ is

If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$,then which one of the following statements is not correct?

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^{2}B^{2} - B^{2}A^{2})X = O$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix,has ....... .

Let $\theta = \frac{\pi}{5}$ and $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$. If $B = A + A^4$,then $\det(B)$

If $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$,then the determinant of $A^2 - 2A$ is