If H = [[2, 3], [5, 7]], find the eigenvalues of H. (2023)

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Question: If H = [[2, 3], [5, 7]], find the eigenvalues of H. (2023)

Options:

Correct Answer: 1, 8

Exam Year: 2023

Solution:

The eigenvalues are found by solving the characteristic equation: det(H - λI) = 0, which gives λ^2 - 9λ + 1 = 0.

If H = [[2, 3], [5, 7]], find the eigenvalues of H. (2023)

If H = [[2, 3], [5, 7]], find the eigenvalues of H. (2023)

Step 1: Write down the matrix H, which is H = [[2, 3], [5, 7]].
Step 2: Identify the identity matrix I of the same size as H. For a 2x2 matrix, I = [[1, 0], [0, 1]].
Step 3: Define λ (lambda) as a variable representing the eigenvalue.
Step 4: Calculate H - λI. This means subtracting λ from the diagonal elements of H: H - λI = [[2 - λ, 3], [5, 7 - λ]].
Step 5: Find the determinant of the matrix H - 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 quadratic equation λ^2 - 9λ + 1 = 0 using the quadratic formula: λ = [9 ± sqrt(9^2 - 4*1*1)] / (2*1).
Step 9: Calculate the discriminant: 9^2 - 4*1*1 = 81 - 4 = 77.
Step 10: Find the two eigenvalues using the quadratic formula: λ = (9 ± sqrt(77)) / 2.

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 a scalar multiple of 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 about the matrix.