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

[pbwf]

Homework Statement

A is an nxn invertible matrix. Show that

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

Homework Equations

det(AB) = det(A)det(B)

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-\lambdaI) = 0, but I cannot see how to form this given the final equation which we need to produce.

 

[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!