For a square matrix $A_{n \times n}$

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

(B) $|\mathrm{A}|=|\operatorname{adj} \mathrm{A}|^{\mathrm{n}-1}$

(C) $\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}|$

(D) $\left|\mathrm{A}^{-1}\right|=\frac{1}{|\mathrm{~A}|}$

If $\begin{vmatrix} -1 & a & a^2 \\ -1 & b & b^2 \\ -1 & c & c^2 \end{vmatrix}^2 = \lambda$ and $a - b = 1$, $b - c = 2$ and $c - a = 3$, then the value of $\lambda$ is

Let $M_{ij}$ and $A_{ij}$ denote respectively minors and co-factors of the element in the $i^{th}$ row and $j^{th}$ column of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 2 & 3 \\ 4 & -1 & 0 \end{bmatrix}$. Then:

(A) $M_{32} = 6$

(B) $M_{23} = 9$

(C) $A_{32} = -6$

(D) $A_{23} = -9$

Choose the correct answer from the options given below:

If A and B are two square symmetric matrices of same order, then AB-BA is

If $P = \begin{bmatrix}0 & 1\\ 1 & 1\end{bmatrix}$ and $Q = \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$, then which of the following are correct?

(A) $P+Q$ is a singular matrix.

(B) $P-Q$ is a symmetric matrix.

(C) $P$ and $Q$ are non-singular matrices.

(D) $(P+Q)(P-Q) \neq P^2 - Q^2$

Choose the correct answer from the options given below:

Given that $Y_{3\times k}$, $W_{n\times 3}$ and $P_{p\times k}$ are matrices of specified order, then the condition for n, p, k so that $3PY + 2WY$ is well defined, is:

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