If $\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 3 & 5 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} m & 14 \\ 2 & n \end{bmatrix}$, then $m + n$ is equal to

For some constant 'k', if the system of linear equations

$2x - y + 3z = 1$

$x - 2y + z = 3$

$kx + y - z = 0$

has a unique solution, then

If $\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $x + y + z$ is

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$. If |adj A| = |A|², then which of the following statements are correct?

(A) A is a skew symmetric matrix.

(B) A is a non-singular matrix.

(C) A is a square matrix of order 4.

(D) A is a symmetric matrix.

Choose the correct answer from the options given below:

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

Let $A = [a_{ij}]$ be a square matrix of order 2 with elements either 0 or 1. Then the difference between the possible number of singular and non-singular matrices is

If A and B are square matrices of same order n, then identify correct statements from the statements given below:

A. $|adj\ A| = |A|^{n-1}$

B. $|A \cdot B| = |B| \cdot |A|$

C. $adj\ A' = (adj\ A)'$

D. $adj\ AB = (adj\ A) \cdot (adj\ B)$

E. $|A^n| = |A|^n$

Choose the correct answer from the options given below: