What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?

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Question: What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?

Options:

Correct Answer: λ^2 - 3λ + 1

Solution:

The characteristic polynomial is given by det(A - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)(2-λ) - 1 = λ^2 - 3λ + 3.

What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?

What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?

Step 1: Identify the matrix A, which is [[2, 1], [1, 2]].
Step 2: Define λ (lambda) as a variable that will be used in the characteristic polynomial.
Step 3: Create the identity matrix I of the same size as A, which is [[1, 0], [0, 1]].
Step 4: Calculate λI, which is [[λ, 0], [0, λ]].
Step 5: Subtract λI from A to get A - λI, resulting in [[2-λ, 1], [1, 2-λ]].
Step 6: Find the determinant of the matrix A - λI, which is det([[2-λ, 1], [1, 2-λ]]).
Step 7: Use the formula for the determinant of a 2x2 matrix: det([[a, b], [c, d]]) = ad - bc.
Step 8: Apply the formula: (2-λ)(2-λ) - (1)(1) = (2-λ)(2-λ) - 1.
Step 9: Expand (2-λ)(2-λ) to get (4 - 4λ + λ^2).
Step 10: Combine the terms: λ^2 - 4λ + 4 - 1 = λ^2 - 4λ + 3.
Step 11: The characteristic polynomial is λ^2 - 4λ + 3.

Characteristic Polynomial – The characteristic polynomial of a matrix is derived from the determinant of the matrix subtracted by λ times the identity matrix.
Determinants – Understanding how to compute the determinant of a 2x2 matrix is essential for finding the characteristic polynomial.
Eigenvalues – The roots of the characteristic polynomial correspond to the eigenvalues of the matrix.