Let $a = \lim_{x \to 1} \left( \frac{x}{\ln x} - \frac{1}{x \ln x} \right)$,$b = \lim_{x \to 0} \frac{x^3 - 16x}{4x + x^2}$,$c = \lim_{x \to 0} \frac{\ln(1 + \sin x)}{x}$,and $d = \lim_{x \to -1} \frac{(x + 1)^3}{3(\sin(x + 1) - (x + 1))}$. Then the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{-1, 0, 1\}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is:

Let $A$ be a $3 \times 3$ matrix such that $X^{T}AX = O$ for all nonzero $3 \times 1$ matrices $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $A \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}1 \\ 4 \\ -5\end{array}\right]$,$A \left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 4 \\ -8\end{array}\right]$,and $\operatorname{det}(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma$,where $\alpha, \beta, \gamma \in \mathbb{N}$,then $\alpha^2+\beta^2+\gamma^2$ is

Let $P=\begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order $3$. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$,then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

$A$ is a $3 \times 3$ matrix satisfying $A^3-5A^2+7A+I=0$. If $A^5-6A^4+12A^3-6A^2+2A+2I=lA+mI$, then $l+m=$

Let $X = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. Let $Y$ be a $2 \times 2$ real matrix satisfying the condition $XY = YX$. Then the smallest possible value of $\det(Y)$ is