Verify A(adj A)=(adj A)A=\(\mid A \mid I\).
\(\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}\)
A=\(\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}\)
IAI=1(0-0)+1(9+2)+2(0-0)=11
IAII=\(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}11&0&0\\0&11&0\\0&0&11\end{bmatrix}\)
Now,A11=0,A12=-(9+2)=-11, A13=0
A21=-(-3-0)=3, A22=3-2=1,A23=-(0+1)=-1
A31=2-0=2, A32=-(-2-6)=8, A33=0+3=3
therefore adj A=[032 -1118 0-13]
Now,A(adjA)= \(\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}\begin{bmatrix}0&3&2\\-11&1&18\\0&-1&3\end{bmatrix}\)
=\(\begin{bmatrix}0+11+0&3-1-2&2-8+6\\0+0+0&9+0+2&6+0-6\\0+0+0&3+0-3&2+0+9\end{bmatrix}\)
=\(\begin{bmatrix}11&0&0\\0&11&0\\0&0&11\end{bmatrix}\)
Also ,(adjA).A=\(\begin{bmatrix}0&3&2\\-11&1&18\\0&-1&3\end{bmatrix}\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}\)
=\(\begin{bmatrix}0+9+2&0+0+0&0-6+6\\-11+3+8&11+0+0&-22-2+24\\0-3+3&0+0+0&0+2+9\end{bmatrix}\)
=\(\begin{bmatrix}11&0&0\\0&11&0\\0&0&11\end{bmatrix}\)
Hence A(adjA)=(adj A)A=IAII.
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