Question

For the following matrix A, find A^-1 if it exists.

Answer 1

Given the matrix A :

`A=\begin{bmatrix}{0} & {0} & {1} \\ {1} & {0} & {0} \\ {0} & {1} & {0}\end{bmatrix}`

The determinant of the matrix will be =

`1\cdot(1\cdot1-0)=1`

Now, we will find the transpose of the matrix :

`A^T=\begin{bmatrix}{0} & {1} & {0} \\ {0} & {0} & {1} \\ {1} & {0} & {0}\end{bmatrix}`

Then, find the elements of the inverse :

`\text{adj(A)}=\begin{bmatrix}{0} & {1} & {0} \\ {0} & {0} & {1} \\ {1} & {0} & {0}\end{bmatrix}`

So, the inverse will be :

`A^(-1)=(adj(A))/(\det (A))=\begin{bmatrix}{0} & {1} & {0} \\ {0} & {0} & {1} \\ {1} & {0} & {0}\end{bmatrix}`