# How do I find the eigenvalue given unknown rows & eigen vect

## Homework Statement

Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x.

A =
4 4 2
* * *
* * *
x =
2
-1
10

Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue?

## The Attempt at a Solution

4−λ 4 2
a b−λ c
d e f−λ=

det (λI3-3) =

−λ3+b+f+4×λ2+4×a−4×b+2×d+c×e−b×f−4×f×λ+−2×b×d+4×c×d+2×a×e−4×c×e−4×a×f+4×b×f

Don't know if I'm on the right track. Trying to find the roots of this characteristic equation, but there's so many unknowns.

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fresh_42
Mentor
What is a eigenvector?

What is a eigenvector?
An eigenvector of A is a non-zero vector x such that Ax = λx

fresh_42
Mentor
So. Have you already computed Ax?

So. Have you already computed Ax?
Yes

24 a2-b+c10 d2-e+f10

Yes

24 a2-b+c10 d2-e+f10

I have 5 minutes left to answer the question. Would you mind helping me with the answer, and then you can explain it to me?

fresh_42
Mentor
Yes

24 a2-b+c10 d2-e+f10
I assume, this should be a column vector. Now what says the equation ##Ax = \lambda x## in the first coordinate?

I assume, this should be a column vector. Now what says the equation ##Ax = \lambda x## in the first coordinate?
Sorry? You mean 24?

I got it! Thanks!

fresh_42
Mentor
I mean the first coordinate is 24 on the left and ##\lambda x_1## on the right, yes.

I mean the first coordinate is 24 on the left and ##\lambda x_1## on the right, yes.
I figured it out, thanks a bunch!