If $A = \begin{bmatrix} 4 & -1 & \lambda \\ 2 & 0 & 6 \\ 1 & -2 & 7 \end{bmatrix}$, then $\frac{1}{2}A^{-1}$ exist if

If $A = \begin{bmatrix} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{bmatrix}$, then $A^2$ is equal to :

If $A = \begin{bmatrix} 2 & -3 & 4 \\ -3 & 5 & x \\ 4 & 3 & 0 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & -10 \\ -2 & z & 6 \\ y & -6 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $(xy + yz + zx)$ is

Let $A = \begin{bmatrix} 1 & 2 & 1 \\ -1 & 3 & 2 \\ 2&4&1\end{bmatrix}$ and $M_{ij}$, $A_{ij}$ respectively denote the minor, co-factor of an element $a_{ij}$ of matrix A, then which of the following are true?

(A) $M_{22} = -1$

(B) $A_{23} = 0$

(C) $A_{32} = 3$

(D) $M_{23} = 1$

(E) $M_{32} = -3$

Choose the correct answer from the options given below:

If $A$ is square matrix of order 3 and $A \cdot (Adj.(A)) = 10I$, then the value of $\frac{1}{25}|Adj.(A)|$ is

The value of $\lambda$ for which the following system of equations has no solution is

$x+y+2z=4$

$2x+y+3z=5$

$x+3y+\lambda z=7$

The minimum value of $\begin{vmatrix}2 & 2 & 2 \\ 2 & 2+x & 2 \\ 2 & 2 & 2+x\end{vmatrix}$, $x \in R$ is