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 :

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