For any 3 ×3 matrix $A$, if $A(adj\,A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then $2|A|$ is equal to

If $f(x) = \begin{vmatrix} 0 & x+a & x+b \\ x-a & 0 & x+c \\ x-b & x-c & 0 \end{vmatrix}$ then the value of $f(0)$ is equal to:

Which of the following statements are correct?

A. If $P$ and $Q$ are square matrices of same order, then $(PQ^T-QP^T)$ is a skew symmetric matrix.

B. The number of all possible matrices of order $3\times2$ with each entry either –1 or 1, is 64.

C. If $P$ is an invertible matrix, then $P^T$ is also invertible.

D. If $P$ and $Q$ are square matrices of the same order, then $|PQ|=|P|+|Q|$.

Choose the correct answer from the options given below:

Which of the following statements are TRUE?

A. If the product of two matrices is a zero matrix, it is not necessary that one of the matrix is a zero matrix.

B. For any three matrices A, B and C, $(AB)C = A(CB)$.

C. Diagonal elements of a skew symmetric matrix can be non-zero.

D. Multiplication of diagonal matrices of same order is always commutative.

Choose the correct answer from the options given below:

Let $A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}$, then adjoint (A) is:

If the matrix $\left[\begin{array}{rrr}0 & -1 & 3 x \\ 1 & y & -5 \\ -6 & 5 & 0\end{array}\right]$ is skew-symmetric, then the value of $5 x-y$ is:

The value of the determinant $\Delta=\begin{vmatrix}\cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha\\-\sin\beta & \cos\beta & 0\\\sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha\end{vmatrix}$ is: