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I have this here matrix

[tex]A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 3 & 3 & 0 \end{array} \right][/tex]

I calculate the eigenvalues and get (2,1,-1)

Next I calculate the eigenvectors and get (1,0,1) and (-1,1,0) and (0,0,0)

My professor says my third eigenvector is wrong and it should (0,0,1)

My calculation:

[tex]A = \left[ \begin{array}{ccc} (2-(-1) & 1 & 0 \\ 0 & (1-(-1) & 0 \\ 3 & 3 & 1-(-1) \end{array} \right]

= \left[ \begin{array}{ccc} 3 & 1 & 0 \\ 0 & 2 & 0 \\ 3 & 3 & 0 \end{array} \right][/tex]

Then according to the theorem regarding eigenvectors:

[tex]\left[ \begin{array}{ccc} 3 & 1 & 0 \\ 0 & 2 & 0 \\ 3 & 3 & 0 \end{array} \right] \left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right] [/tex]

then

[tex]3v_1 + v_2 = 0[/tex]

[tex] 2v_2 = 0[/tex]

[tex]3v_1 + 3 v_2 = 0[/tex]

Is my calculations correct ??

sincerley and best regards,

Fred