Help with multiplicity & eigenvalues...

A is matrix:
[5 0 4]
[5 1 5]
[-3 0 -2]

Here is my work to find the eigenvalues (using x instead of lambda)
[ 5-x 0 4]
[5 x-1 5]
-3 0 -2-x]

(x - 1) ( x^2 - 3x + 2)
(x - 1) (x - 2) (x - 1)

Eigenvalues of 1 and 2
Multiplicity of 1 and 0

My eigenvalues are correct (I think) but both multiplicities are incorrect????

I thought multiplicity referred to the number of times the eigenvalue appears in the matrix but obviously I am wrong.. Please help!

You're not quite correct. The multiplicity of an eigenvalue is the number of times it appears as a root to the characteristic equation.
In other words, you have (x-1)(x-2)(x-1)
So 1 is a root of two of the factors, which means that the multiplicity of the eigenvalue 1 is two.
Also, 2 is a root of a single factor, meaning the multiplicity of the eigenvalue 2 is one.

Hope that clears it up.

The eigenvalue of 1 has a multiplicity of 2 (because there's 2 of them) The eigenvalue of 2 has multiplicity 1.