Guess Eigenvalue of 2x2 Matrix with Constant k Sum

[fk378]

Homework Statement

Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A?

a. [1,0]
b. [0,1]
c. [1,1]

(The answer can be any or all of these)

The Attempt at a Solution

I don't know how to approach this...

 

[tiny-tim]

Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvalue of A?

a. [1,0]
b. [0,1]
c. [1,1]

(The answer can be any or all of these)
…
I don't know how to approach this...

Hi fk378!

Sorry to say this, but … just go for the screamingly obvious :

for a 2x2 matrix ab cd, what is its effect on each of the three given vectors?

 

[sutupidmath]

a. [1,0]
b. [0,1]
c. [1,1]

...

...and here i thought that eigenvalues were scalars

 

[HallsofIvy]

Yes, it is. However, fk378 did ask "Which of the following must be an eigenvector of A?
Presumably, after determining which of those vectors is an eigevector, you could then determine the corresponding eigenvalue. That's certainly better than "guessing" the eigenvalue!

fk378, what is
\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}?
What is
\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix}?
What is
\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 1\end{bmatrix}?

Remembering, of course that a+ c= k, b+ d= k, a+ b= k, and c+ d= k.

 

[fk378]

AH sorry! I had a typo in the subject line...it should read "guess the eigenvector"

So sorry for the confusion here.

 

[sutupidmath]

Yes, it is. However, fk378 did ask "Which of the following must be an eigenvector of A?
.

Well, the OP defenitely edited it at some latter point, since when i looked it it read eigenvalue.

 

[tiny-tim]

I wasn't confused

anyway, what's your answer …

for a 2x2 matrix ab cd, what is its effect on each of the three given vectors?