If $\begin{vmatrix}2a & 2a & 3b+c\\3b & 2a+c & 3b\\2a+3b & c & c\end{vmatrix} = 48$, then abc is equal to

Assume $P$, $Q$, $R$ and $S$ are matrices of order $2 \times m$, $k \times n$, $m \times 2$ and $2 \times 3$ respectively. The restrictions on $k$, $m$ and $n$, so that $PQ + RS$ is defined are

For a matrix $A = \begin{bmatrix} a & -1 & b \\ -2 & 1 & -1 \\ 1 & 0 & 2 \end{bmatrix}$, $C_{32} = -1$ and $C_{33} = 3$, where $C_{ij}$ is the cofactor of element $a_{ij}$ in $A$. Then

Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true?

(A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

(B) det $(A) = 5$

(C) det (adjA) = 25

(D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$

Choose the correct answer from the options given below:

If A is a square matrix of order 3, B = kA and |B| = $x$|A| then,

The matrix $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ is a :

(A) scalar matrix

(B) diagonal matrix

(C) skew-symmetric matix

(D) symmetric matrix

Choose the correct answer from the options given below :

If A is a square matrix such that $A^2 = I$, then $(A+I)^3 + (A-I)^3 - 7A$ is equal to