Question

Prove that the following matrix is orthogonal & hence find 𝑨−𝟏.

A`=1/3[(-2,1,2),(2,2,1),(1,-2,2)]`

Answer 1

Let A`=1/3[(-2,1,2),(2,2,1),(1,-2,2)]`

Transpose of A is given by ,

`A^T=1/3[(-2,2,1),(1,2,-2),(2,1,2)]`

`A.A^T=1/3[(-2,1,2),(2,2,1),(1,-2,2)][(-2,2,1),(1,2,-2),(2,1,2)]`

`=1/9[(1,0,0),(0,1,0),(0,0,1)]`

`=1/9`

The given matrix A is orthogonal.
The inverse of an orthogonal matrix is always equal to the
Transpose of that particular matrix.

`therefore A^-1=1/3[(-2,2,1),(1,2,-2),(2,1,2)]`