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

[Razberryz]

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?

Homework Equations

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.

 

[fresh_42]

What is a eigenvector?

 

[Razberryz]

What is a eigenvector?

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

 

[fresh_42]

So. Have you already computed Ax?

 

[Razberryz]

So. Have you already computed Ax?

Yes

24 a2-b+c10 d2-e+f10

 

[Razberryz]

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]

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?

 

[Razberryz]

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

Sorry? You mean 24?

 

[Razberryz]

I got it! Thanks!

 

[fresh_42]

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

 

[Razberryz]

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!