If a matrix $A=\begin{bmatrix} 3 & -5 & 7 \\ 5 & 2 & 4 \\ -7 & -4 & 9 \end{bmatrix}$ is expressed as the sum of a symmetric matrix P and a skew symmetric matrix Q, then symmetric matrix P will be

The matrix $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ is

Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

If the matrix $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$, then the value of det $(A^2 - 2A)$ is equal to

If A is a square matrix such that $A^2=A$ and I is the identify matrix of the same order as A then $(I + 2A)^3$-6A is equal to

The number of all possible matrices of order 3 with each entry either 0 or 1 is:

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$. If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to