# Finding a Fundamental Matrix for the System

## Homework Statement

Find a fundamental matrix for the system x'(t)=Ax(t); where
5 -3 -2
A = 8 -5 -4
-4 3 3

The second part of the question is to find eAt

## Homework Equations

I know that you have to find the 3 eigenvectors and then the 'general solution' without putting the constants into the fundamental matrix. I'm having a super hard time finding the last eigen vector.

## The Attempt at a Solution

I found the eigenvalues to be (λ-1)3 and when I completed for the first λ=1 I found that v1=[1 4 0] and v2= [1 0 2]. I tried to generalize to find v3 by using (A-λI)v3=v2 but I keep getting an invalid system.

Any help would be great.

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## Answers and Replies

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Mark44
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## Homework Statement

Find a fundamental matrix for the system x'(t)=Ax(t); where
5 -3 -2
A = 8 -5 -4
-4 3 3

The second part of the question is to find eAt

## Homework Equations

I know that you have to find the 3 eigenvectors and then the 'general solution' without putting the constants into the fundamental matrix. I'm having a super hard time finding the last eigen vector.

## The Attempt at a Solution

I found the eigenvalues to be (λ-1)3
No, the eigenvalue is λ = 1.
and when I completed for the first λ=1 I found that v1=[1 4 0] and v2= [1 0 2].
You made a mistake in v1 - it isn't an eigenvector. You can check this by verifying that Av1 ≠ 1v1.

Your other eigenvector is correct.
I tried to generalize to find v3 by using (A-λI)v3=v2 but I keep getting an invalid system.

Any help would be great.

Dick