If $A = \begin{bmatrix}\alpha & \beta\\r & \Delta\end{bmatrix}$, such that $|A| = 25$, then the value of $\left|\frac{1}{5}A\right|$ is.

If $\begin{bmatrix} -1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is a singular matrix, then the relation between $a$ and $b$ is:

The system of equations

$x - 3y - 8z = -10$

$2x + 5y + \lambda z = 13$

$3x + y - 4z = 0$

has infinite number of solutions if the value of $\lambda$ is equal to:

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 the system of equations

$x + 2y + 3z = 10$

$-x + y + \lambda z = 20$

$2x + 3y + \lambda z = 0$

does not possess a unique solution, then $\lambda$ is equal to

For the matrix $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$, which of the following statements are correct?

(A) The order of the matrix is 3 × 3

(B) |A| = 21

(C) $|adj\ A| = 225$

(D) A is skew symmetric matrix

Choose the correct answer from the options given below:

If $A^T = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then the matrix $(A + 2B)^T$ is