If $a$, $b$ and $c$ are distinct prime numbers then the value of $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$ is equal to

Let A, B be square matrices of order $3\times3$. Then $|A.adj(A).B|$ is

If P, Q and R are matrices of order 2x3, 3x5 and 5x3 respectively. Then which of the following are valid?

(A) P Q R

(B) P R Q

(C) Q R

(D) R Q

(E) P R

Choose the correct answer from the options given below:

Match List–I with List–II

Choose the correct answer from the options given below

If A is a square matrix of order 4 and $|adj\,A| = 8$, then $|A| =$

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

The matrix X for which $\begin{bmatrix}5 & 4\\1 & 1\end{bmatrix}X=\begin{bmatrix}1 & -2\\1 & 3\end{bmatrix}$ is