If $A$,$B$,and $C$ are square matrices of order $3$ such that $A = \begin{bmatrix} x & 0 & 1 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ and $|B| = 36$,$|C| = 4$,$(x, y, z \in \mathbb{N})$ and $|ABC| = 1152$,then the minimum value of $x + y + z$ is

The value of $\theta$ lying between $\theta = 0$ and $\theta = \pi / 2$ and satisfying the equation : $\left| \begin{array}{ccc} 1 + \sin^2 \theta & \cos^2 \theta & 4 \sin 4 \theta \\ \sin^2 \theta & 1 + \cos^2 \theta & 4 \sin 4 \theta \\ \sin^2 \theta & \cos^2 \theta & 1 + 4 \sin 4 \theta \end{array} \right| = 0$ are :

If $A(\theta)=\begin{bmatrix} i \sin \theta & \cos \theta \\ \cos \theta & i \sin \theta \end{bmatrix}$ is a matrix,where $i=\sqrt{-1}$,then which of the following is not true?

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

Let $A = \begin{bmatrix} p & 13 \\ -13 & p \end{bmatrix}$ and $B = \begin{bmatrix} 4q & 85 \\ -2 & 1 \end{bmatrix}$ where $p, q \in N$. It is given that $|A| = |B|$ and $p, q \in [1, 1000]$. Then the total number of ordered pairs $(p, q)$ is:

If $D_1$ and $D_2$ are two $3 \times 3$ diagonal matrices,then