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## Homework Statement

A linear transformation with Matrix A = ##

\begin{pmatrix}

5&4&2\\

4&5&2\\

2&2&2

\end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1.

## Homework Equations

3. The Attempt at a Solution [/B]

I know from the trace of matrix A that eigenvalue 1 has a multiplicity of 2 and that eigenvalue 10 has a multiplicity of 1. Solving for eigenvectors corresponding for ## \lambda = 1 ##, pretend it's an augmented matrix...

##

\begin{pmatrix}

4&4&2\\

4&4&2\\

2&2&1

\end{pmatrix}

R_2 ->R_2 - R_1;

R_3 ->R_3 - \dfrac{1}{2} \cdot R_2 =

\begin{pmatrix}

4&4&2\\

0&0&0\\

0&0&0

\end{pmatrix}

R_1-> \dfrac{1}{2} \cdot R_1 =

\begin{pmatrix}

2&2&1\\

0&0&0\\

0&0&0

\end{pmatrix}

##

Therefore ##x_2## and ##x_3## are arbitrary. Solving for ##x_1## gives $$x_1=-x_2-\dfrac{1}{2} x_3$$ $$x_2=x_2$$ $$x_3=x_3$$

$$v= \begin{pmatrix}

-x_2-\dfrac{1}{2} x_3\\

x_2\\

x_3

\end{pmatrix} $$

##v= x_2 \begin{pmatrix}

-1 \\

1 \\

0

\end{pmatrix} +

x_3 \begin{pmatrix}

-1 \\

0 \\

2

\end{pmatrix} ## Now those are the two eigenvectors I got however the answer in my book says the two eigenvectors are ##v_1=<1,0,-2>## and ##v_2=<0,1,-2>##