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**1. The problem statement, all variables and given/known data**

Which of the following is

**not**an eigenvector for [tex]

T \left(

\left[ {\begin{array}{cc}

x \\

y \\

\end{array} } \right] \right) =

\left[ {\begin{array}{cc}

x + y \\

x+ y \\

\end{array} } \right]

[/tex] ?

A) v = [-2 -2]

^{T}

B) v = [1 -1]

^{T}

C) v = [1 2]

^{T}

D) All are eigenvectors

**2. Relevant equations**

Ax = [tex]\lambda[/tex]x

**3. The attempt at a solution**

My problem is that all eigenvectors I've computed have come from 2x2 matrices. My best guess on starting is

[tex]

T \left(

\left[ {\begin{array}{cc}

-2 \\

-2 \\

\end{array} } \right] \right) =

\left[ {\begin{array}{cc}

-4 \\

-4 \\

\end{array} } \right]

\left[ {\begin{array}{cc}

-2 \\

-2 \\

\end{array} } \right]

=

[/tex]

but this obviously doesn't work because of the size. How do I find an eigenvalue of a 2x1 matrix? Is it possible? Am I even looking at this correctly?

Edit: I think I've figured it out. First, I constructed the standard matrix for T and got [tex]\left[ {\begin{array}{cc}

1&1 \\

1&1 \\

\end{array} } \right] [/tex]

Then I used Ax = [tex]\lambda[/tex]x with the vectors given to find the eigenvalues. Letter C didn't have an eigenvalue, so that is the answer.

[tex]

T \left(

\left[ {\begin{array}{cc}

-2 \\

-2 \\

\end{array} } \right] \right) =

\left[ {\begin{array}{cc}

1&1 \\

1&1 \\

\end{array} } \right]

\left[ {\begin{array}{cc}

-2 \\

-2 \\

\end{array} } \right] =

\left[ {\begin{array}{cc}

-4 \\

-4 \\

\end{array} } \right] = 2

\left[ {\begin{array}{cc}

-2 \\

-2 \\

\end{array} } \right]

[/tex]

[tex]

T \left(

\left[ {\begin{array}{cc}

1 \\

-1 \\

\end{array} } \right] \right) =

\left[ {\begin{array}{cc}

1&1 \\

1&1 \\

\end{array} } \right]

\left[ {\begin{array}{cc}

1 \\

-1 \\

\end{array} } \right] =

\left[ {\begin{array}{cc}

0 \\

0 \\

\end{array} } \right] = 0

\left[ {\begin{array}{cc}

1 \\

-1 \\

\end{array} } \right]

[/tex]

[tex]

T \left(

\left[ {\begin{array}{cc}

1 \\

2 \\

\end{array} } \right] \right) =

\left[ {\begin{array}{cc}

1&1 \\

1&1 \\

\end{array} } \right]

\left[ {\begin{array}{cc}

1 \\

2 \\

\end{array} } \right] =

\left[ {\begin{array}{cc}

3 \\

3 \\

\end{array} } \right] = ???

[/tex]

Last edited: