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

• Nov 5th 2009, 03:18 PM
flabbergastedman
Proof help (related to characteristic polynomials and Cayley-Hamilton)
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:

\$\displaystyle 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.
• Nov 6th 2009, 04:13 AM
tonio
Quote:

Originally Posted by flabbergastedman
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:

\$\displaystyle 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 \$\displaystyle a_0=(-1)^ndet(A)\$

Tonio
• Nov 6th 2009, 12:18 PM
flabbergastedman
ugh I must have been falling asleep when I was looking at this problem. I just remembered that fact, especially given that \$\displaystyle a_o\$ is not equal to 0. Thank you!