Question

Find the inverse of each of the matrices (if it exists). \(\begin{bmatrix}1&0&0\\3&3&0\\5&2&-1\end{bmatrix}\)

Answer 1

Let A=\(\begin{bmatrix}1&0&0\\3&3&0\\5&2&-1\end{bmatrix}\)

we have IAI=1(-3-0)-0+0=-3
now A₁₁=-3-0=-3, A₁₂=-(-3-0)=3, A₁₃=6-15=9
A₂₁=-(0-0)=0, A₂₂=-1-0=-1,A₂₃=-(2-0)=-2
A₃₁=0-0=0,A₃₂=-(0-0)=0,A₃₃=3-0=3

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

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