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This is the matrix A, which i need to find the eigenvalues and eigenvectors.

3x3 matrix

5 6 12

0 2 0

-1 -2 -2

The attempt at a solution

I have found the eigenvalues to be: 1, 2, 2.

So, the final eigenvalues are : 1 and 2.

Now, i found the eigenvector for eigenvalue = 1, which is:

3x1 column matrix:

[-3 0 1]^T

But for the eigenvalue = 2, i am stuck, as these are the system equations that i have before me:

3x1 + 6x2 + 12x3 = 0

-x1 - 2x2 - 4x3 = 0

I made x1 the subject of formula: -2x2 - 4x3

And then i'm not sure how to proceed. But i'm going out on a limb here, so please correct me.

Let x2 = 1 and x3 = 0

Then i get this 3x1 column matrix:

x2[-2 1 0]^T

Let x3 = 1 and x2 = 0

I get another 3x1 column matrix:

x3[-4 0 1]^T

So, all the eigenvectors in a 3x3 matrix P, are:

-3 -2 -4

0 1 0

1 0 1

Is this correct?? Most importantly, is my method correct? Is there a better method?

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# Find the eigenvectors given the eigenvalues

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