Question

If $A=\left[\begin{array}{c}23\\ 12\end{array}\right],B=\left[\begin{array}{c}10\\ 31\end{array}\right],$ find $AB$ and ${\left(AB\right)}^{-1}.$ verify that ${\left(AB\right)}^{-1}$=$B^{-1}{A}^{-1}$.

Answer 1

$A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right],B-\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right]|A|=|B|=1$

$AB=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right]=\left[\begin{array}{cc}11& 3\\ 7& 2\end{array}\right]|AB|=22-21=1$

$(AB{)}^{-1}=\frac{adj\left(AB\right)}{|AB|}=adj(AB)=\left[\begin{array}{cc}2& -3\\ -7& 11\end{array}\right]$

$B^{-1}=\frac{adj\left(B\right)}{|B|}=adj\left(B\right)=\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]$

$A^{-1}=\frac{adj\left(A\right)}{|A|}=adj\left(A\right)=\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

$B^{-1}{A}^{-1}=\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}2& -3\\ -7& 11\end{array}\right]$

$(AB{)}^{-1}=B^{-1}{A}^{-1}$