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

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

Correct Answer: 1 and 3

Step 1: Write down the matrix A, which is [[2, 1], [1, 2]].
Step 2: Define λ (lambda) as a variable that will help us find the eigenvalues.
Step 3: Create the identity matrix I of the same size as A, which is [[1, 0], [0, 1]].
Step 4: Calculate λI, which is [[λ, 0], [0, λ]].
Step 5: Subtract λI from A to get A - λI, which results in [[2-λ, 1], [1, 2-λ]].
Step 6: Find the determinant of the matrix A - λI. The determinant is calculated as (2-λ)(2-λ) - (1)(1).
Step 7: Simplify the determinant expression: (2-λ)(2-λ) - 1 = (2-λ)^2 - 1.
Step 8: Expand (2-λ)^2 to get 4 - 4λ + λ^2, then subtract 1 to get λ^2 - 4λ + 3.
Step 9: Set the determinant equal to zero: λ^2 - 4λ + 3 = 0.
Step 10: Factor the quadratic equation: (λ - 1)(λ - 3) = 0.
Step 11: Solve for λ to find the eigenvalues: λ = 1 and λ = 3.

Eigenvalues – Eigenvalues are scalars associated with a matrix that indicate how much the eigenvectors are stretched or compressed during a linear transformation.
Characteristic Polynomial – The characteristic polynomial is derived from the determinant of the matrix subtracted by λ times the identity matrix, and its roots give the eigenvalues.
Determinants – The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties about the matrix, including whether it is invertible.