Finding matrices with a given eigenvector/eigenvalue

Problem: Find all 2x2 matrices for which http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D is an eigenvector with associated eigenvalue -1.

Here's my attempt:

http://latex.codecogs.com/gif.latex?...0%5Cvec%7Bv%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

This is where I start to get off track, assuming I was actually on the correct track. A push in the right direction would be great.

Thanks.

Quote:

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Originally Posted by tangibleLime

Problem: Find all 2x2 matrices for which http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D is an eigenvector with associated eigenvalue -1.

Here's my attempt:

http://latex.codecogs.com/gif.latex?...0%5Cvec%7Bv%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

http://latex.codecogs.com/gif.latex?...d%7Bbmatrix%7D

This is where I start to get off track, assuming I was actually on the correct track. A push in the right direction would be great.

Thanks.

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One hundred percent correct, actually! Just solve those last two simultaneous equations!

2a+3b=-2

b=-(2a+2)/3

2c+3d=-3

d=-(2c+3)/3

http://quicklatex.com/cache3/ql_f5e3...40befc2_l3.png

Huzzah! Thanks, I got it.

And thanks to alexmahone for confirming my answer.