6.1#8

Hi Professor Taylor I've been stuck on this problem for a while now and I keep getting that the basis of the eigenvalue of -4 should be [0,1,0,0] for the matrix
|0,0,0,0|
|0,0,-4,-4|
|-4,0,2,2|
|-4,0,-2,6|
I'm not sure if I'm doing my math wrong or not. Thank you for your time.

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Well, you've computed that A-(-4)Identity correctly, and you've guessed that [0,1,0,0] is in the null space. But you're also tying yourself in knots, because the eigenvectors of -4 will be all vectors in the null space of  A-(-4)Identity, which you can solve easily by doing row reduction on  A-(-4)Identity. (Hint: there's more than one vector in a basis for this null space)