Question

If A is a `2xx2` matrix such that `A^(2)-4A+3I=O`, then prove that `(A+3I)^(-1)=7/24 I-1/24 A`.

Answer 1

We have, `A^(2)-4A+3I=O`
`implies A(A+3I)-7A+3I=O`
`implies A(A+3I)-7(A+3I)=-24 I`
`implies (A+3I) (A-7I)=-24 I`
`implies (A+3I) (7/24 I-A/24)=I`
`implies (A+3 I)^(-1) =7/24 I- A/24`