Identify statements $S_{\mathit{1}}\mathit{,}\mathit{ }{S}_{\mathit{2}}\mathit{,}\mathit{ }{S}_{\mathit{3}}$ in order for true$\left(\mathrm{T}\right)$/false$\left(\mathrm{F}\right)$

$S_1:$ if $A=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right],$ then $\mathrm{adj} A=A\text{'}$

$S_2:$ if $A=\left[\begin{array}{lll}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right],$ then $A^{-1}=\left[\begin{array}{lll}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$

$S_3:$ If $B$ is a non-singular matrix and $A$ is a square matrix, then $det\left(B^{-1}AB\right)=det\left(A\right)$

STATEMENT-1: If $A=\left[\begin{array}{lll}{a}^2+x^2& ab-cx& ac+bx\\ ab+xc& b^2+x^2& bc-ax\\ ac-bx& bc+ax& c^2+x^2\end{array}\right]$ and $B=\left[\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right],$ then $\left|A\right|={\left|B\right|}^2.$.

STATEMENT-2: If $A^c$ is cofactor matrix of a square matrix $A$ of order $n$, then $\left|A^c\right|=A^{n\mathit{-}\mathit{1}}.$

Let $\lambda$ and $\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations have infinite solution $\forall$ real values of $\alpha .$

$\lambda x+\left(\sin \alpha y\right)+\left(\cos \alpha z\right)=0$

$x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0$

$-x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0$