# Show formula of inverse matrix

Let

p(Y)=(-Y)^n + a_(n-1)*Y^(n-1) + ... + a_0

be the characteristic polynomial of matrix A. Show that A is invertible, if and only if a_0 isn't zero and that inverse of A is

A^(-1)=q(A),

where q is polynomial.

So i'm having trouble showing that inverse is polynomial of A.

For the first part i went this way; A is invertible when its determinant isnt zero. p(Y)=det(A-YI), so p(0)=det(A), so det(A)=a_0.

For inverse i dont even know how to start. Any guidance is much appreciated!

Hint: Multiply Cayley-Hamilton relation $$A^n + a_{n-1} A^{n-1} + \cdots + a_1 A + a_0 = 0$$ by $A^{-1}.$