I am having problems trying to orthogonally diagonalise the below matrix. I am stuck at the stage where you find the eigenvalues.

The matrix is

5 -4 0

-4 3 -4 = 0

0 -4 1

This is what I get, but I cannot solve it so I think I have done something wrong.

5-λ -4 0

-4 3-λ -4 = 0

0 -4 1-λ

(5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0

this works down to

(5-λ)(λ^2 +12λ - 29)

when typing this I have just realised that I just need to use the quadratic equation to solve (λ^2 +12λ - 29) and (5 - λ) is just λ = 5.

So if someone could just confirm if I'm heading towards the right direction or not, then that would be great

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