What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?
Correct Answer: λ^2 - 5λ + 2
- Step 1: Identify the matrix A. Here, A = [[1, 2], [3, 4]].
- Step 2: Define λ (lambda) as a variable. We will use it to find the characteristic polynomial.
- Step 3: Create the identity matrix I of the same size as A. For a 2x2 matrix, I = [[1, 0], [0, 1]].
- Step 4: Calculate A - λI. This means subtracting λ times the identity matrix from A: A - λI = [[1-λ, 2], [3, 4-λ]].
- Step 5: Write down the determinant formula for a 2x2 matrix. The determinant of [[a, b], [c, d]] is ad - bc.
- Step 6: Apply the determinant formula to our matrix: det([[1-λ, 2], [3, 4-λ]]) = (1-λ)(4-λ) - (2*3).
- Step 7: Simplify the expression: (1-λ)(4-λ) = 4 - 4λ - λ + λ^2 = λ^2 - 5λ + 4. Then subtract 6: λ^2 - 5λ + 4 - 6 = λ^2 - 5λ + 2.
- Step 8: The characteristic polynomial is λ^2 - 5λ + 2.
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