Question

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

Answer 1

Let A=\(\begin{bmatrix}1&-1&2\\0&2&-3\\3&-2&4\end{bmatrix}\)
By expanding along C₁,we have :
IAI=1(8-6)-0+3(3-4)=2-3=-1
Now A₁₁=8-6=2, A₁₂=-(0+9)=-9, A₁₃=0-6=-6
A₂₁=-(-4+4)=0, A₂₂=4-6=-2, A₂₃=-(-2+3)=-1
A₃₁=3-4=-1, A₃₂=-(-3-0)=3, A₃₃=2-0=2

so adj A=\(\begin{bmatrix}2&0&-1\\-9&-2&3\\-6&-1&2\end{bmatrix}\)

so A^-1=\(\frac{1}{\mid A \mid}\)adj A=- \(\begin{bmatrix}2&0&-1\\-9&-2&3\\-6&-1&2\end{bmatrix}\)

=\(\begin{bmatrix}-2&0&1\\9&2&-3\\6&1&-2\end{bmatrix}\)