find eigenvalues

answer
det(A-λ I) =

3-λ........0 .........2.......... .0
1...........3-λ........1 .........0
0............1 .........1-λ .......0
0 ...........0 ..........0 ........4-λ
=(4-λ)*
3-λ........0 .........2
1...........3-λ........1
0............1 .........1-λ
=(4-λ)[(3-λ)^2(1-λ)+2-(3-λ)]
=(4-λ)[(3-λ)[(3-λ)(1-λ)-1]+2]
=(4-λ)[(3-λ)[λ^2-4λ+2]+2]
=(4-λ)(-λ^3+7λ^2-14λ+8)
=(λ-4)(λ^3-7λ^2+14λ-8)
Let (λ-4)(λ^3-7λ^2+14λ-8)=0
λ=4 or λ^3-7λ^2+14λ-8=0
since λ^3-7λ^2+14λ-8=0
(λ-2)(λ^2-5λ+4)=0
(λ-2)(λ-4)(λ-1)=0
λ=1 or λ=2 or λ=4
So the eigenvalues of matrix A are 1,2,4 (repeated)