Compute $(AB{)}^{-1}$ when $A=\left[\begin{array}{ccc}1& 1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]$and $B^{-1}=\left[\begin{array}{ccc}1& 2& 0\\ 0& 3& -1\\ 1& 0& 2\end{array}\right]$.

Obtain the inverses of the matrices $\left[\begin{array}{ccc}1& \mathrm{p}& 0\\ 0& 1& \mathrm{p}\\ 0& 0& 1\end{array}\right]$ and $\left[\begin{array}{lll}1& 0& 0\\ \mathrm{q}& 1& 0\\ 0& 9& 1\end{array}\right]$. And, hence find the inverse of the matrix $=\left[\begin{array}{ccc}1+pq& p& 0\\ q& 1+pq& p\\ 0& q& 1\end{array}\right]$

If $A=\left[\begin{array}{ll}3& 2\\ 2& 1\end{array}\right]$, verify that $A^2–4A–I=O$, and hence find $A^{-1}$.

Show that the matrix $A=\left[\begin{array}{cc}-8& 5\\ 2& 4\end{array}\right]$, satisfies the equation $x^2+4x-42=0$, and hence find $A^{-1}$

If $A=\left[\begin{array}{cc}-1& -1\\ 2& -2\end{array}\right]$, show that ${\mathrm{A}}^2+3\mathrm{A}+4{\mathrm{I}}_2=\mathrm{O}$ and hence find $A^{-1}$

If $A=\left[\begin{array}{ll}3& 1\\ 7& 5\end{array}\right]$, find $x$ and $y$ such that $A^2+xI=yA$. Hence, find $A^{-1}$

If $A=\left[\begin{array}{ll}3& -2\\ 4& -2\end{array}\right]$. Find the value of $\lambda$ so that $A^2=\lambda A–2I$. Hence, find $A^{-1}.$

Show that the $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& -1& 2\\ 3& 4& 1\end{array}\right]$ satisfies the equation $A^3–A^2–3A–I=O$, and hence find $A^{-1}.$

Prove that: $adj I=I$