Proof help (related to characteristic polynomials and Cayley-Hamilton)

flabbergastedman · #1 $Old$ November 5th, 2009, 04:18 PM

I'm a bit stuck on where to start on this proof. The problem reads:

"Let A be an n x n matrix with characteristic polynomial:

$f(t) = (-1)^nt^n + a(n-1)t^{n-1} + ... + a(1)t + a(0)$

Prove that A is invertible iff a(0) is not equal to Zero.

The (n-1), (1) and (0) are subscripts, my apologies I don't have/know how to use Latex to form the equations appropriately.

tonio · #2 $Old$ November 6th, 2009, 05:13 AM

Quote:

Originally Posted by flabbergastedman $View Post$

I'm a bit stuck on where to start on this proof. The problem reads:

"Let A be an n x n matrix with characteristic polynomial:

$f(t) = (-1)^nt^n + a(n-1)t^{n-1} + ... + a(1)t + a(0)$

Prove that A is invertible iff a(0) is not equal to Zero.

The (n-1), (1) and (0) are subscripts, my apologies I don't have/know how to use Latex to form the equations appropriately.

Well, just prove that $a_0=(-1)^ndet(A)$

Tonio

flabbergastedman · #3 $Old$ November 6th, 2009, 01:18 PM

ugh I must have been falling asleep when I was looking at this problem. I just remembered that fact, especially given that $a_o$ is not equal to 0. Thank you!