Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, find adj (AB).

Answer 1

$$\begin{gathered} A\times B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right] \\ A\times B\mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{singular}\mathrm{matrix}.\mathrm{Therefore},\mathrm{it}\mathrm{is}\mathrm{invertible}. \\ \mathrm{Let}{C}_{ij}\mathrm{be}\mathrm{a}\mathrm{cofactor}\mathrm{of}{a}_{ij}\mathrm{in}A. \\ \mathrm{Th}e\mathrm{cofactors}\mathrm{of}\mathrm{element}A\mathrm{are}\mathrm{given}\mathrm{by} \\ {C}_{11}=d \\ {C}_{12}=-c \\ {C}_{21}=-b \\ {C}_{22}=a \\ \therefore \mathrm{adj}A={\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]}^T=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right] \end{gathered}$$