Find the eigenvalues of the matrix G = [[5, 4], [2, 3]]. (2020)

Find the eigenvalues of the matrix G = [[5, 4], [2, 3]]. (2020)

Step 1: Write down the matrix G, which is [[5, 4], [2, 3]].
Step 2: Identify the identity matrix I of the same size as G, which is [[1, 0], [0, 1]].
Step 3: Multiply the identity matrix I by a variable λ (lambda), resulting in [[λ, 0], [0, λ]].
Step 4: Subtract λI from G to get the matrix G - λI, which is [[5 - λ, 4], [2, 3 - λ]].
Step 5: Calculate the determinant of the matrix G - λI. The determinant is calculated as (5 - λ)(3 - λ) - (4)(2).
Step 6: Expand the determinant: (5 - λ)(3 - λ) = 15 - 5λ - 3λ + λ^2 = λ^2 - 8λ + 15. Then subtract 8: λ^2 - 8λ + 15 - 8 = λ^2 - 8λ + 7.
Step 7: Set the determinant equal to zero to form the characteristic equation: λ^2 - 8λ + 7 = 0.
Step 8: Factor the characteristic equation: (λ - 1)(λ - 7) = 0.
Step 9: Solve for λ by setting each factor equal to zero: λ - 1 = 0 gives λ = 1, and λ - 7 = 0 gives λ = 7.
Step 10: Conclude that the eigenvalues of the matrix G are λ = 1 and λ = 7.

Eigenvalues – Eigenvalues are scalars associated with a matrix that provide insights into its properties, found by solving the characteristic polynomial.
Characteristic Equation – The characteristic equation is derived from the determinant of the matrix subtracted by 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, used to determine if a matrix is invertible and to find eigenvalues.