Find the eigenvalues of A = [[2, 1], [1, 2]].

Step 1: Recall the characteristic equation.

The eigenvalues λ satisfy:
det(A - λI) = 0

Step 2: Form A - λI.

A = [[2, 1], [1, 2]]

Subtract λ from the diagonal entries:

A - λI = [[2 - λ, 1], [1, 2 - λ]]

Step 3: Find the determinant.

det(A - λI) = (2 - λ)(2 - λ) - (1)(1)

So:
(2 - λ)^2 - 1 = 0

Step 4: Expand the expression.

(2 - λ)^2 - 1 = 4 - 4λ + λ^2 - 1

Therefore:
λ^2 - 4λ + 3 = 0

Step 5: Factor the characteristic polynomial.

λ^2 - 4λ + 3 = (λ - 1)(λ - 3)

Step 6: Set each factor equal to zero.

λ - 1 = 0 gives λ = 1.

λ - 3 = 0 gives λ = 3.

Therefore, the eigenvalues are 1 and 3.