# Linear Algebra - Proof of the characteristic polynomial for the inverse of A

1. Nov 23, 2009

### pbwf

1. The problem statement, all variables and given/known data

A is an nxn invertible matrix. Show that

PA-1(x) = ($$\frac{(-x)^n}{det(A)}$$) *PA(1/X)

2. Relevant equations
det(AB) = det(A)det(B)

3. The attempt at a solution

Ok from what I understand this is the proof for the characteristic polynomial for the inverse of a matrix. The professor said that we need to use somehow the equation above.

I also understand that the equation is usually of the form det(A-$$\lambda$$I) = 0, but I cannot see how to form this given the final equation which we need to produce.

2. Nov 22, 2010

### yourself_15

Try substituting the definition of the characteristic polynomial into the equation you are trying to verify. You get $$\det(Ix- A^{-1}) = \frac{(-x)^n\det(Ix^{-1} - A)}{\det(A)}$$ If you notice that $$\det(-Ix) = (-x)^n$$, then the rest is matrix arithmetic using the equation you provided.

Good luck!