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Mathematics Question on Determinants

Find the inverse of each of the matrices(if it exists). [213\4−10−721]\begin{bmatrix}2&1&3\\\4&-1&0\\\\-7&2&1\end{bmatrix}

Answer

Let A=[213\4−10−721]\begin{bmatrix}2&1&3\\\4&-1&0\\\\-7&2&1\end{bmatrix}
We have IAI=2(-1-0)-1(4-0)+3(8-7)
=-2-4+3
=-3
Now A11=-1-0=-1, A12=-(4-0), A13=8-7=1
A21=-(1-6)=5, A22=2+21=23, A23=-(-(4+7)=-11
A31=0+3=3, A32=-(0-12)=12,A33=-2-4=-6

therefore adj A=[−153−42312\1−11−6]\begin{bmatrix}-1&5&3\\\\-4&23&12\\\1&-11&-6\end{bmatrix}

so A-1=1∣A∣\frac{1}{\mid A\mid}adj A=−13-\frac{1}{3} [−153−42312\1−11−6]\begin{bmatrix}-1&5&3\\\\-4&23&12\\\1&-11&-6\end{bmatrix}