Find the eigenvalues of the matrix G = [[2, 1], [1, 2]]. (2020)

Find the eigenvalues of the matrix G = [[2, 1], [1, 2]]. (2020)

Step 1: Write down the matrix G = [[2, 1], [1, 2]].
Step 2: Define λ (lambda) as a variable that represents the eigenvalue.
Step 3: Create the identity matrix I of the same size as G, which is I = [[1, 0], [0, 1]].
Step 4: Calculate G - λI. This means subtracting λ from the diagonal elements of G: G - λI = [[2-λ, 1], [1, 2-λ]].
Step 5: Find the determinant of the matrix G - λI. The determinant is calculated as: det(G - λI) = (2-λ)(2-λ) - (1)(1).
Step 6: Simplify the determinant expression: det(G - λI) = (2-λ)(2-λ) - 1 = (2-λ)^2 - 1.
Step 7: Expand the expression: (2-λ)(2-λ) = 4 - 4λ + λ^2, so det(G - λI) = λ^2 - 4λ + 3.
Step 8: Set the determinant equal to zero to find the characteristic polynomial: λ^2 - 4λ + 3 = 0.
Step 9: Factor the polynomial: (λ - 1)(λ - 3) = 0.
Step 10: Solve for λ: The solutions are λ = 1 and λ = 3, which are the eigenvalues.

Eigenvalues – Eigenvalues are scalars associated with a matrix that provide insights into its properties, found by solving the characteristic polynomial.
Characteristic Polynomial – The characteristic polynomial is derived from the determinant of the matrix subtracted by λ times the identity matrix, used to find eigenvalues.
Determinants – Determinants are scalar values that can be computed from the elements of a square matrix, important for finding eigenvalues.