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-$$\lambda$$I) = 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!