Eigenvalu problem

Homework Statement
[PLAIN]http://img14.imageshack.us/img14/7826/70745131.jpg [Broken]

The attempt at a solution
How do I go about solving this problem?

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2..
is dis correct!

2..
is dis correct!

????

Mark44
Mentor
Maybe, maybe not.

Mark44
Mentor
What does it mean to say that a number lambda is an eigenvalue of a matrix?

it means that no. is the eigen value of new matrix formed by the expression..
A^2010 -2A+3I

if for an eigen vector,eigen value is 1...dis means for that e-vector, matrix is behaving like an identity matrix...so the same eigen vector this expression will have eigen value :2

How did you show it was 2?

Mark44
Mentor
it means that no. is the eigen value of new matrix formed by the expression..
A^2010 -2A+3I
You didn't answer my question. Here's a slightly different question. What does it mean to say that 3 is an eigenvalue of a matrix A?

matrix can act in two ways ...rotating a vector or changing it length...
this value...(e-value) is a factor by which matrix changes the length of vector...

You didn't answer my question. Here's a slightly different question. What does it mean to say that 3 is an eigenvalue of a matrix A?

Mark44
Mentor
I'm looking for an equation that involves A and 3.

I'm looking for an equation that involves A and 3.

Is it:

(A - 3I)x = 0

Mark44
Mentor
Yes, or equivalently, Ax = 3x.

With your problem, you know that 1 is an eigenvalue of A, so Ax = 1x. You are trying to find an eigenvalue of A2010 - 2A + 3I.

(A2010 - 2A + 3I)x = ?x

Mark44
Mentor
Think about the other problem you posted. If Ax = 1x, what are A2x, A3x, A4x, ...?

All would be 1x

Mark44
Mentor
OK, so what would (A2010 - 2A + 3I)x be?

Also 1x

Mark44
Mentor
Also 1x
No it isn't.

No it isn't.

Whoops lol, I meant 2.

So would it be enough to say that since A=1, A^2010 - 2A + 3I = 1 -2 + 3 = 2?

Mark44
Mentor
So would it be enough to say that since A=1, A^2010 - 2A + 3I = 1 -2 + 3 = 2?
No, A is a matrix, so it can't be equal to any number, and A^2010 - 2A + 3I $\neq$ 2

However, you know that Ax = 1x, so (A2010 - 2A + 3I)x = ___x? (Fill in the blank.)