We are given a square matrix A and the equal-sized identity matrix i. We are also given that n > 0 is the smallest whole integer for which real coefficients a_0 up until a_(n-1) exist, so that the following equation is valid:

A^n + a_(n-1)A^(n-1) + ... + a_1A + a_0 i = 0

Display that A is invertible if and only if a_0 =/= 0.

I have no idea which theorem I can use to solve this problem. If anyone could help me get started, that’d be great. We basically just learned about eigenvalues, eigenvectors, etc. So I assume that I have to do something with that, but I just don’t seem to he able to connect any of the theorems to this problem. Could anyone help?

EDIT: I managed to solve the problem by making the assumption that a_0 does, in fact, equal 0 and finding a proof by contradiction.

A^n + a_(n-1)A^(n-1) + ... + a_1A + a_0 i = 0

Display that A is invertible if and only if a_0 =/= 0.

I have no idea which theorem I can use to solve this problem. If anyone could help me get started, that’d be great. We basically just learned about eigenvalues, eigenvectors, etc. So I assume that I have to do something with that, but I just don’t seem to he able to connect any of the theorems to this problem. Could anyone help?

EDIT: I managed to solve the problem by making the assumption that a_0 does, in fact, equal 0 and finding a proof by contradiction.

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