Let $B=\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$ and $A$ be a $2 \times 2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$,then $2\beta-\alpha$ is equal to:

Let $X = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}$. If $f: X \rightarrow \mathbb{R}$ is defined by $f(A) = \det(A)$ for all $A \in X$, then $f$ is

The number of cubic polynomials $P(x)$ satisfying $P(1)=2, P(2)=4, P(3)=6, P(4)=8$ is

Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

Let $a, b, c$ be non-real numbers satisfying the equation $x^5 = 1$ and $S$ be the set of all non-invertible matrices of the form $\begin{bmatrix} 1 & a & b \\ w & 1 & c \\ w^2 & w & 1 \end{bmatrix}$,where $w = e^{\frac{i 2\pi}{5}}$. Then the number of distinct matrices in the set $S$ is:

If the minimum and the maximum values of the function $f : \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$,defined by $f(\theta) = \left|\begin{array}{ccc} -\sin^2 \theta & -1-\sin^2 \theta & 1 \\ -\cos^2 \theta & -1-\cos^2 \theta & 1 \\ 12 & 10 & -2 \end{array}\right|$ are $m$ and $M$ respectively,then the ordered pair $(m, M)$ is equal to: