Can someone kindly help me find the Eigenvector for this 2x 2 matrix

Hi,

Here is the question

1) Find the characteristic equation, eigvenvalues and their eigenvectors for
A = [ 3 0 ]
......[ 5 4 ]

I got the charateristic eequation, eigenvalues but i cant get the eigenvectors.. its too complicated for me

So please help me
thank you

I trust you got λ = 3 and λ = 4 for your eigenvalues.

When λ = 3, A - λI is given by

[ 0..0 ]
[ 5..1 ]

Let u =

[u1]
[u2]

be an eigenvector associated with λ = 3. Then (A - λI)u = 0 implies

5u1 + u2 = 0

So u2 = -5u1. Choose u1 = 1, for example; then u2 = -5. If you want a unit vector, divide the components by the length of u (which is √26).

Do the same thing for λ = 4; you will find that u1 = 0, so u2 = 1 gives a unit vector.