Proof of an inverse Matrix

Question

I am very raw at proofs, this is only my first semester learning them and I am having trouble with this problem. How would I approach this ?

If $$ is an $ × $ matrix that satisfies the equation $ $ is $A^3 -4A^2 +3A -5I_n = 0$, Find the $A^{-1}$

Is there anymore propety as to $A$ such as nilponent or idemponet? — Not a Salmon Fish, Feb 03 '21 at 06:56
Hint: Write your equation as $$A\Big(\frac{A^2-4A+3I}{5}\Big)=I$$ — Matthew H., Feb 03 '21 at 07:08

PTDS · Accepted Answer · 2021-02-03T07:54:45.230

2

$$A^3 -4A^2 +3A -5I_n = 0$$

Or, $$5I_n = A^3 -4A^2 +3A$$

Pre-multiplying both sides by $A^{-1}$ and dividing by $5$, $$A^{-1} = \frac{1}{5}(A^2 -4A +3I_n)$$

edited Feb 03 '21 at 07:54

answered Feb 03 '21 at 07:14

PTDS

1 Answers1