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If A is a 2xx2 matrix such that A^(2)-4A...

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`.

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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`
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CENGAGE ENGLISH-MATRICES-CAE 13.5
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