Inverse of a Matrix

Consider the following statements in respect of square matrices $A, B, C$ each of same order $n$ :

$1.$ $AB=AC\Rightarrow B=C$ if $A$ is nonsingular

$2.$ If $BX=CX$ for every column matrix $X$ having $n$ rows then $B=C$

Which of the statements given above is/are correct?

If $S_k=\left(\begin{array}{ll}1& k\\ 0& 1\end{array}\right), k\in N,$ where $N$ is the set of natural numbers, then ${\left(S_2\right)}^n{\left(S_k\right)}^{-1}, \forall n\in N$ is equal to

If $B=\left[\begin{array}{cc}-2& -2\\ -1& 0\end{array}\right]$ and $A$ is a matrix such that $A^{-1}B=B^{-1}$ and $k{A}^{-1}=2B^{-1}+I,$ where $k$ is some scalar and $I$ is the $2\times 2$ identity matrix, then the value of $k$ is

If the inverse of the matrix $A=\left[\begin{array}{rrr}1& 2& 2\\ 2& -1& 2\\ 2& 2& 1\end{array}\right]$ is $\frac{1}{5}\left[\begin{array}{ccc}-3& 2& 2\\ 2& -3& \alpha \\ 2& 2& -3\end{array}\right]$ then, $\alpha =$

If $Q$ is a non-singular matrix, then the value of $adj\left(Q^{-1}\right)$ in terms of $Q$ is

If $A=\left[\begin{array}{cc}-2& 4\\ 1& -3\end{array}\right]$, then $A^{-1}=$.

If $A^T{A}^{-1}$ is symmetric, then $A^2=$

If ${\left(AB\right)}^{-1}=\left[\begin{array}{cc}12& -17\\ -19& 27\end{array}\right]$ and $A^{-1}=\left[\begin{array}{cc}1& -1\\ -2& 3\end{array}\right]$ then $B^{-1}=$

If $A=\left[\begin{array}{ll}7& 3\\ 4& 2\end{array}\right]$, then $9I_2-A=$

If $A$ is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}A$ and $B=A^{-1}{A}^T$, then $B{B}^T=$

Let $A={\left(a_{ij}\right)}_{1\le i, j\le 3}$ be a $3\times 3$ invertible matrix where each $a_{ij}$ is a real number. Denote the inverse of the matrix $A$ by $A^{-1}.$ If $\sum _{j=1}^3{a}_{ij}=1$ for $1\le i\le 3,$ then

Let $A$ be a $2\times 2$ matrix of the form $A=\left[\begin{array}{ll}a& b\\ 1& 1\end{array}\right],$ where $a, b$ are integers and $-50\le b\le 50.$ The number of such matrices $A$ such that $A^{-1},$ the inverse of $A,$ exists and $A^{-1}$ contains only integer entries is

If $A=\left[\begin{array}{ll}3& 4\\ 1& 2\end{array}\right]$, then $A^{-1}$ is equal to