Solveeit Logo
Login

Question

Mathematics Question on Determinants

Verify A(adj A)=(adj A)A=IAII. [23−4−6]\begin{bmatrix}2&3\\\\-4&-6\end{bmatrix}

Answer

A=[23−4−6]\begin{bmatrix}2&3\\\\-4&-6\end{bmatrix}
we have IAI=-12-(-12)=-12+12=0

so IAII=0∣10\01∣\begin{vmatrix}1&0\\\0&1\end{vmatrix}=∣00\00∣\begin{vmatrix}0&0\\\0&0\end{vmatrix}

Now A11=-6, A12=4, A21=-3, A22=2

so adj A=∣−63\42∣\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}

=∣−12+12−6+6\24−2412−12∣\begin{vmatrix}-12+12&-6+6\\\24-24&12-12\end{vmatrix}=∣00\00∣\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}

=∣−12+12−18+18\8−812−12∣\begin{vmatrix}-12+12&-18+18\\\8-8&12-12\end{vmatrix}

=∣00\00∣\begin{vmatrix}0&0\\\0&0\end{vmatrix}

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