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

In summary, the conversation is about proving the characteristic polynomial for the inverse of a matrix. The equation det(AB) = det(A)det(B) is mentioned and the suggestion is to substitute the definition of the characteristic polynomial into the given equation. This results in the equation \det(Ix- A^{-1}) = \frac{(-x)^n\det(Ix^{-1} - A)}{\det(A)}, which can be simplified using the fact that \det(-Ix) = (-x)^n. The rest involves matrix arithmetic.
  • #1
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Homework Statement



A is an nxn invertible matrix. Show that

PA-1(x) = ([tex]\frac{(-x)^n}{det(A)}[/tex]) *PA(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-[tex]\lambda[/tex]I) = 0, but I cannot see how to form this given the final equation which we need to produce.
 
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  • #2
Try substituting the definition of the characteristic polynomial into the equation you are trying to verify. You get [tex] \det(Ix- A^{-1}) = \frac{(-x)^n\det(Ix^{-1} - A)}{\det(A)} [/tex] If you notice that [tex] \det(-Ix) = (-x)^n [/tex], then the rest is matrix arithmetic using the equation you provided.

Good luck!
 

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

1. What is the characteristic polynomial for the inverse of A?

The characteristic polynomial for the inverse of A is the polynomial that is used to calculate the eigenvalues of the inverse matrix. It is denoted by p(x) and is given by p(x) = det(xI - A), where I is the identity matrix and A is the original matrix.

2. Why is it important to prove the characteristic polynomial for the inverse of A?

Proving the characteristic polynomial for the inverse of A is important because it provides a way to calculate the eigenvalues of the inverse matrix, which can be useful in solving a variety of problems in linear algebra. It also helps to understand the properties and behavior of the inverse matrix.

3. How is the characteristic polynomial for the inverse of A derived?

The characteristic polynomial for the inverse of A is derived by first expanding the determinant of the matrix xI - A using Laplace expansion. Then, the properties of determinants are used to simplify the expression and obtain the characteristic polynomial.

4. Can the characteristic polynomial for the inverse of A be used to find the eigenvalues of A?

No, the characteristic polynomial for the inverse of A cannot be used to directly find the eigenvalues of A. However, it can be used to find the eigenvalues of the inverse matrix, which can then be used to find the eigenvalues of A using the relationship between eigenvalues of a matrix and its inverse.

5. Are there any alternative methods to prove the characteristic polynomial for the inverse of A?

Yes, there are alternative methods to prove the characteristic polynomial for the inverse of A, such as using the Cayley-Hamilton theorem or the Jordan canonical form. However, the method of expanding the determinant and using determinants properties is the most commonly used and straightforward approach.

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