For what values of $\lambda$ and $\mu$, the following system of linear equations:

$x+y+2z=1, 2x+y-z=4, x+2y+\lambda z=\mu$

have infinitely many solution?

If $A = \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}$, then which of the following statements are TRUE?

[where $A_{ij}$ is the cofactor of the (i, j)th element, $a_{ij}$ of matrix A, $1 \gt i, j \le 3$.]

(A) $a_{11}A_{31} = -24$

(B) $a_{12}A_{32} = -66$

(C) $a_{13}A_{33} = -90$

(D) $a_{21}A_{21} = -24$

Choose the correct answer from the options given below:

If A is square matrix of order 3 x 3 such that $a_{ij} = j - i$, then which of the following statements are correct?

(A) A is a symmetric matrix.

(B) A is a skew symmetric matrix.

(C) $|A| = 0$.

(D) A is a diagonal matrix.

Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ 3 & -4 \\ 2 & 4 \end{bmatrix}$ then product AB is :

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then $A^{26}$ is equal to

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A)

(A) $(AB)^{-1} = B^{-1} A^{-1}$

(B) $(A+B)' = A' \times B'$

(C) $(AB)' = A'B'$

(D) $|kA| = k^3 |A|$

(E) $|A'| = |A|$

Choose the correct answer from the options given below :

If $A$ is a non-singular matrix of order $n$, then $adj(adj\ A)$ is equal to :