Characteristic polynomial
This is the determinant of
```
 [1–λ 2 ] 
 [ 0 3–λ]
```
which is (1–λ)(3–λ)–2(0)=(1–λ)(3–λ). I don't want to multiply things, because the next step uses the factored form.
Eigenvalues
The roots of the characteristic polynomial are 1 and 3 and these are the eigenvalues.
Bases for eigenspaces
- λ=1
  So
```
 [1–λ 2 ]
 [ 0 3–λ]
```
  becomes
```
 [0 2] 
 [0 2]
```
  and the vectors which solve the corresponding homogeneous system all have their second coordinates equal to 0. A basis for that subspace is the set consisting of the single vector
```
[1]
[0].
```
- λ=3
  So
```
 [1–λ 2 ]
 [ 0 3–λ]
```
  becomes
```
 [–2 2] 
 [ 0 0]
```
  and the vectors which solve the corresponding homogeneous system all have the difference of their coordinates equal to 0 (using the first row: –2x₁+2x₂=0, so x₁=x₂). A basis for that subspace is the set consisting of the single vector
```
[1]
[1].
```

The matrix can can be diagonalized: A=PDP^–1 where P is

 [1 1]
 [0 1]

and D is

 [1 0]
 [0 3].