Question

If A is a square matrix such that A^2=2I,then A^-1=?​

Answer 1

`A^2=2I`

Recall that
`I=AA^(-1)=A^(-1)A`, so

`A^2=2(A^(-1)A)`

Multiply on the right by
`A^(-1)` and regroup terms:

`A^2A^(-1)=2(A^(-1)A)A^(-1)`

`A(AA^(-1))=2A^(-1)(AA^(-1))`

`A=2A^(-1)`

and finally solve for
`A^(-1)`:

`\boxed{A^(-1)=\frac12A}`