Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-$3 . If the points $(x, y)$ satisfying $\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $\mathrm{e}$ and the length of the latus rectum is $l$, then $\mathrm{e}^4+l^4$ is equal to ________.

Let A be a 3 $\times$ 3 matrix such that $\mathrm{|adj(adj(adj~A))|=12^4}$. Then $\mathrm{|A^{-1}~adj~A|}$ is equal to

If the system of equations $x+y+a z=b$ $2 x+5 y+2 z=6$ $x+2 y+3 z=3$ has infinitely many solutions, then $2 a+3 b$ is equal to :

Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for some $$n \in…

If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$,…

Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where $$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr…