Question

Verify A(adj A)=(adj A)A=IAII. \(\begin{bmatrix}2&3\\-4&-6\end{bmatrix}\)

Answer 1

A=\(\begin{bmatrix}2&3\\-4&-6\end{bmatrix}\)
we have IAI=-12-(-12)=-12+12=0

so IAII=0\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=\(\begin{vmatrix}0&0\\0&0\end{vmatrix}\)

Now A¹¹=-6, A¹²=4, A²¹=-3, A²²=2

so adj A=\(\begin{vmatrix}-6&3\\4&2\end{vmatrix}\)

Now A(adj A)=\(\begin{bmatrix}2&3\\-4&-6\end{bmatrix}\)\(\begin{vmatrix}-6&3\\4&2\end{vmatrix}\)

=\(\begin{vmatrix}-12+12&-6+6\\24-24&12-12\end{vmatrix}\)=\(\begin{vmatrix}0&0\\0&0\end{vmatrix}\)

Also (adj A)A=\(\begin{vmatrix}-6&3\\4&2\end{vmatrix}\)\(\begin{bmatrix}2&3\\-4&-6\end{bmatrix}\)

=\(\begin{vmatrix}-12+12&-18+18\\8-8&12-12\end{vmatrix}\)

=\(\begin{vmatrix}0&0\\0&0\end{vmatrix}\)

Hence A(adj A)=(adj A)A=IAII.