If a matrix has 12 elements, then the possible orders it can have, are

(A) 1 × 12

(B) 4 × 3

(C) 6 × 3

(D) 6 × 2

Choose the correct answer from the options given below:

Identify the correct statements.

(A) If $A$ is a non-singular matrix, then $A^{-1} = \frac{|A|}{(\text{adj } A)}$

(B) If $A$ is an invertible matrix then $\frac{1}{|A^{-1}|} = |A|$

(C) If $A$ and $B$ are two invertible matrices of the same order then $AB$ is also invertible matrix and $(BA)^{-1} = A^{-1}B^{-1}$

(D) If $A$ is an invertible matrix, then $A^T$ is also invertible and $(A^T)^{-1} = \frac{1}{(A^{-1})^T}$

Choose the correct answer from the options given below :

Let the matrix $A = [a_{ij}]_{3\times3}$ be defined by

$a_{ij} = \begin{cases} 2i + 3j, & i < j \\ 5, & i = j \\ 3i - 2j, & i > j \end{cases}$

The number of elements in the matrix A which are greater than 7, is:

Which of the following statement are correct?

(A) $A = [a_{ij}]_{n \times n}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$

(B) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$

(C) A square matrix $A = [a_{ij}]$ is called a skew-symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$

(D) For every square matrix $A$, there exist an identity matrix of the same order such that $IA = AI = I$

Choose the correct answer from the options given below:

If the matrix $\begin{bmatrix} 0 & 7 & -12 \\ -7 & 0 & -5 \\ 2a & 5 & 3b \end{bmatrix}$ is skew-symmetric, then the value of $(4a + 3b)$ is:

If the system of linear equations

$x + 2y - 3z = 1$

$(2p+1)y + z = 2$

$3x + 3z = 5$

has a unique solution, then p can not be equal to

For some constant 'k', if the system of linear equations

$2x - y + 3z = 1$

$x - 2y + z = 3$

$kx + y - z = 0$

has a unique solution, then