(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given p(x) = x^{4}+2x^{2}+1 and

A =[[1 1 -2 0][0 1 0 2][1 1 -1 1][0 0 -2 -1]]p(A) = 0

Find a polynomial q(x) so that q(A) = A^{-1}

a) What is q(x)?

b) Compute q(A) = A^{-1}

2. Relevant equations

I found the Cayley-Hamilton theorem, which states: p(x) = det(A-xI_{n}).

3. The attempt at a solution

I found the inverse of A, which is:

[[-1 -1 2 0][4 1 -4 -2][1 0 -1 -1][-2 0 2 1]]From here I thought I could set q(x) = A^{-1}.

[[-1-x -1 2 0][4 1-x -4 -2][1 0 -1-x -1][-2 0 2 1-x]]Then I should just solve out to find the det(). I used the 2nd column as a reference point.

Doing this I end up with the equation, x^{4}-6x^{2}+1.

so q(x) = x^{4}-6x^{2}+1.

This seems to be the wrong answer. I'm not at all sure if I did this right, or if I made an error in my calculations. Would really appreciate if someone could look over and point out my error.

Thanks

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# Linear Algebra - polynomial functions of matrices

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