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What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?

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Question: What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?

Options:

  1. λ^2 - 5λ - 2
  2. λ^2 - 5λ + 2
  3. λ^2 + 5λ + 2
  4. λ^2 + 5λ - 2

Correct Answer: λ^2 - 5λ + 2

Solution: 

The characteristic polynomial is det(A - λI) = det([[1-λ, 2], [3, 4-λ]]) = (1-λ)(4-λ) - 6 = λ^2 - 5λ + 2.

What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?

Practice Questions

Q1
What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?
  1. λ^2 - 5λ - 2
  2. λ^2 - 5λ + 2
  3. λ^2 + 5λ + 2
  4. λ^2 + 5λ - 2

Questions & Step-by-Step Solutions

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.
  • 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.
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