If G = [[2, 3], [5, 7]], find the eigenvalues of G.

If G = [[2, 3], [5, 7]], find the eigenvalues of G.

Step 1: Identify the matrix G, which is given as G = [[2, 3], [5, 7]].
Step 2: Define λ (lambda) as a variable that represents the eigenvalue we want to find.
Step 3: Create the identity matrix I of the same size as G. For a 2x2 matrix, I = [[1, 0], [0, 1]].
Step 4: Calculate G - λI. This means subtracting λ from the diagonal elements of G: G - λI = [[2-λ, 3], [5, 7-λ]].
Step 5: Find the determinant of the matrix G - λI. The determinant is calculated as (2-λ)(7-λ) - (3)(5).
Step 6: Simplify the determinant expression: (2-λ)(7-λ) - 15 = λ^2 - 9λ - 1.
Step 7: Set the determinant equal to zero to form the characteristic equation: λ^2 - 9λ - 1 = 0.
Step 8: Solve the characteristic equation using the quadratic formula: λ = [9 ± sqrt(9^2 - 4*1*(-1))] / (2*1).
Step 9: Calculate the discriminant: 9^2 - 4*(-1) = 81 + 4 = 85.
Step 10: Find the two eigenvalues using the quadratic formula: λ = (9 ± sqrt(85)) / 2.
Step 11: Approximate the values of λ to find the eigenvalues, which are approximately λ = 1 and λ = 8.

Eigenvalues – Eigenvalues are scalars associated with a linear transformation represented by a matrix, found by solving the characteristic equation.
Characteristic Equation – The characteristic equation is derived from the determinant of the matrix minus λ times the identity matrix, set to zero.
Determinant – The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix.