If A = [[1, 2], [3, 4]], what is the eigenvalue of A?

If A = [[1, 2], [3, 4]], what is the eigenvalue of A?

Correct Answer: 5

Step 1: Write down the matrix A, which is [[1, 2], [3, 4]].
Step 2: To find the eigenvalues, we need to calculate the characteristic polynomial. This is done by finding the determinant of (A - λI), where I is the identity matrix and λ is a variable.
Step 3: The identity matrix I for a 2x2 matrix is [[1, 0], [0, 1]].
Step 4: Subtract λI from A: A - λI = [[1-λ, 2], [3, 4-λ]].
Step 5: Now, calculate the determinant of the matrix (A - λI): det([[1-λ, 2], [3, 4-λ]]).
Step 6: The determinant is calculated as (1-λ)(4-λ) - (2)(3).
Step 7: Expand this expression: (1-λ)(4-λ) = 4 - 4λ - λ + λ^2 = λ^2 - 5λ + 4.
Step 8: Now, subtract 6 from this result: λ^2 - 5λ + 4 - 6 = λ^2 - 5λ - 2.
Step 9: Set the characteristic polynomial equal to zero: λ^2 - 5λ - 2 = 0.
Step 10: Solve this quadratic equation using the quadratic formula: λ = [5 ± sqrt(25 + 8)] / 2.
Step 11: Calculate the discriminant: 25 + 8 = 33, so λ = [5 ± sqrt(33)] / 2.
Step 12: The eigenvalues are λ = (5 + sqrt(33))/2 and λ = (5 - sqrt(33))/2.

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