Let $S = \{\sqrt{n} : 1 \leq n \leq 50, n \text{ is odd}\}$. Let $a \in S$ and $A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}$. If $\sum_{a \in S} \operatorname{det}(\operatorname{adj} A) = 100 \lambda$,then $\lambda$ is equal to:

Let $A$ be the set of all $3 \times 3$ matrices with entries $0$ or $1$ only. Let $B$ be the subset of $A$ consisting of all matrices with determinant value $1$. Let $C$ be the subset of $A$ consisting of all matrices with determinant value $-1$. Then:

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 $A^2+xA+yI=0$ lie on a hyperbola,whose transverse axis is parallel to the $x$-axis,eccentricity is $e$ and the length of the latus rectum is $\ell$,then $e^4+\ell^4$ is equal to?

If the determinant of a $3^{\text{rd}}$ order matrix $A$ is $K$,then the sum of the determinants of the matrices $(AA^T)$ and $(A-A^T)$ is

If $p, q, r, s$ are in $A.P.$ and $f(x) = \left| \begin{array}{ccc} p + \sin x & q + \sin x & p - r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s - q + \sin x \end{array} \right|$ such that $\int_{0}^{\pi} f(x) dx = -4$,then the common difference of the $A.P.$ can be:

If $A$ and $B$ are two square matrices such that $B = -A^{-1}BA$,then $(A + B)^2 = $