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:

Let A = PQ. The elementary operation on A, that produces the same effect as it does on applying on P and keeping Q unchanged is :

(A) $R_i \leftrightarrow R_j$

(B) $R_i \to R_i + KR_j$

(C) $C_i \to KC_i$

(D) $C_i \to C_i + KC_j$

Choose the correct answer from the options given below :

If $2 \begin{bmatrix} a & d \\ b & c \end{bmatrix} + 3 \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3 \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$, then the value of $|a + b - c - d|$ is

Let $A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}$, where $0 \leq \theta \leq 2\pi$. Then which of the following are true?

(A) $|A| = 2 + 2 \sin^{2} \theta$

(B) $|A| = 2 + \sin^{2} \theta$

(C) minimum value of $|A|$ is $1$

(D) maximum value of $|A|$ is $4$

Choose the correct answer from the options given below:

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 $\begin{vmatrix}4x & 9\\ -4 & -5\end{vmatrix}=\begin{vmatrix}4 & 8\\ 2 & 3\end{vmatrix}$, then the value of $x$ is :

If $[1 \quad 2 \quad 1] \begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & -1 \\ 0 & 1 & 3 \end{bmatrix} \begin{bmatrix} 2x \\ x \\ -7 \end{bmatrix} = [0]$, then the value of $x$ is