Matrix polynomials and inverses- Linear Algebra

[lina29]

Homework Statement

For p(x)=x⁴-2x³+3x²-3x+1 and

A= 1 1 1 -1
-1 0 -2 1
0 0 1 0
1 0 0 0

you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A^-1. The point is A⁴-2A³+3A²-3A=A(-A³+2A²-3A+3I)=I

a) What is q(x)?

I don't really understand how to approach this problem. My initial though was that I had to solve the right side of the eqn(A⁴-2A³+3A²-3A) and that would be q(x). Am I on the right track? Also what's the purpose of p(x)?

 

[HallsofIvy]

Well, you say, or were given, that "the point is $A(-A^3+2A^2- 3A+ 3I)= I$

Isn't the definition of "inverse" that $AA^{-1}= I$?

 

[Mark44]

Homework Statement

For p(x)=x⁴-2x³+3x²-3x+1 and

A= 1 1 1 -1
-1 0 -2 1
0 0 1 0
1 0 0 0

you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A^-1. The point is A⁴-2A³+3A²-3A=A(-A³+2A²-3A+3I)=I

a) What is q(x)?

I don't really understand how to approach this problem. My initial though was that I had to solve the right side of the eqn(A⁴-2A³+3A²-3A) and that would be q(x). Am I on the right track? Also what's the purpose of p(x)?

Try to be more consistent with your letters. The functions you're working with are named p and q, so you shouldn't be using P in place of p. p(x) = x⁴-2x³+3x²-3x+1, so p(A) = A⁴-2A³+3A²-3A+I. Don't worry about the purpose of p - just take it as given for now.

You have A(-A³+2A²-3A+3I) = I. If A times whatever is the identity, then the whatever is the inverse of A. q(A) will be the inverse of A.