Question

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

Answer 1

Let A=\(\begin{bmatrix}2&1&3\\4&-1&0\\-7&2&1\end{bmatrix}\)
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=\(\begin{bmatrix}-1&5&3\\-4&23&12\\1&-11&-6\end{bmatrix}\)

so A^-1=\(\frac{1}{\mid A\mid}\)adj A=\(-\frac{1}{3}\) \(\begin{bmatrix}-1&5&3\\-4&23&12\\1&-11&-6\end{bmatrix}\)