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Scalar Multiplication of a Matrix

Show that if ...

Question

Show that if a square matrix A satisfies A^2−3A+I=O, then A−1=3I−A.

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Solution

If $A^{2} - 3 A + I = 0$ then assuming $A^{- 1}$ exists multiplying both sides of equation by it
$A^{- 1} (A^{2} - 3 A + I) = A^{- 1} O = O$
Expand brackets
$A - 3 I + A^{- 1} = O A^{- 1} = 3 I - A$

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