Let $\begin{vmatrix} 3 & y \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$ and $x, y$ are natural numbers, then the number of solutions for the system is:

Let A be a square matrix of order n, then which of the following are TRUE?

(A) $|adj A| = |A|^{n-1}$

(B) $|A. adj A| = |A|^n$

(C) $A. (adj A) = |A|$

(D) $|KA| = K|A|$

(E) $|A^{-1}| = \frac{1}{|A|}, |A| \neq 0$

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

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that $A^{-1} = KA$, then the value of K is:

Let A and B be 3×3 matrices such that $A \neq B$. If $A^3 = B^3$ and $A^2B = B^2A$, then the determinant of $A^2 + B^2$ is:

If A and B are square matrices of order 3 such that $|A| = 2$, $|B| = 3$, and $|2A \cdot \text{adj}(3(\text{adj}B))| = 2^\alpha \cdot 3^\beta$, then value of $\alpha + \beta$ is:

If a matrix has 7 elements, then it can be a

(A) Scalar matrix

(B) Symmetric matrix

(C) Column matrix

(D) Row matrix

Choose the correct answer from the options given below: