For a square matrix $A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 4 \\ 1 & 2 & 0 \end{bmatrix}$, which of the following statements is/are TRUE ?

A. det (A) = 9.

B. $A_{21}.A_{33} = -12$, where $A_{ij}$ is the cofactor of $a_{ij}$

C. det (A) = $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{31}$.

D. $\lvert AA'\rvert \neq \lvert A\rvert^2$

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If A and B are two invertible matrices, then which of the following statements are correct?

(A) $|A^{-1}| = |A|^{-1}$

(B) $adjA = |A|A^{-1}$

(C) $(AB)^{-1} = A^{-1}B^{-1}$

(D) $(A + B)^{-1} = A^{-1} + B^{-1}$

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Match List–I with List–II

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If $A$ is a square matrix and $I$ is an identity matrix of same order such that $A^2 = A$, then $(2I + A)^2 - 5A$ is

If $A = \begin{bmatrix} a & 4 & -5 \\ d & b & -6 \\ 5 & e & c \end{bmatrix}$ is a skew symmetric matrix, then value of $a + b + c + d + e$ is equal to

The system of equations

$x + ky = 0$

$3x + 5y = 0$

has infinitely many solutions, then k is equal to

If the matrix $A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, then $A^2$ is equal to: