Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, verify that A(adj A)=(adj A)A=|A|I.

Answer 1

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\Rightarrow \left|A\right|=(4\times 1)-(2\times 3)=4-6=-2C_{ij}={(-1)}^{i+j}{M}_{ij}{C}_{11}={(-1)}^{1+1}\left(4\right)=4C_{12}={(-1)}^{1+2}\left(3\right)=-3C_{21}={(-1)}^{2+1}\left(2\right)=-2C_{22}={(-1)}^{2+2}\left(1\right)=1\therefore C=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]=\left[\begin{array}{cc}4& 3\\ -2& 1\end{array}\right]adj\left(A\right)=C^T=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]A.adj\left(A\right)=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]A.adj\left(A\right)=\left[\begin{array}{cc}4\times 1+2\times -3& -2\times 1+2\times 1\\ 3\times 4-3\times 4& 3\times -2+4\times 1\end{array}\right]A.adj\left(A\right)=\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]=-2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left|A\right|.I(\because -2=\left|A\right|)adj\left(A\right).A=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]=-2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left|A\right|.I\therefore A.adj\left(A\right)=adj\left(A\right).A=\left|A\right|.Iverified$