Cayley Hamilton Theorem Calculator

Enter the elements of a 2×2 matrix into the calculator to determine the characteristic polynomial using the Cayley-Hamilton theorem.

• Matrix Nullity Calculator
• Matrix Transpose Calculator
• Matrix Determinant Calculator (2×2)

Cayley-Hamilton Theorem Formula

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. For a 2×2 matrix, the characteristic polynomial is calculated using the following formula:

λ² - (trace(A))λ + det(A)

Variables:

• λ represents the eigenvalues of the matrix.
• trace(A) is the sum of the elements on the main diagonal of the matrix A.
• det(A) is the determinant of the matrix A.

To calculate the characteristic polynomial for a 2×2 matrix using the Cayley-Hamilton theorem, follow these steps:

1. First, determine the trace of the matrix, which is the sum of the elements a₁₁ and a₂₂.
2. Next, calculate the determinant of the matrix, which is (a₁₁ * a₂₂) – (a₁₂ * a₂₁).
3. Use the formula λ² – (trace(A))λ + det(A) to find the characteristic polynomial.
4. Finally, use the calculator above to verify your result.

Example Problem:

Use the following matrix as an example problem to test your knowledge.

Matrix A:

a₁₁ = 3, a₁₂ = 2

a₂₁ = 1, a₂₂ = 4

Trace(A) = a₁₁ + a₂₂ = 3 + 4 = 7

Det(A) = (a₁₁ * a₂₂) – (a₁₂ * a₂₁) = (3 * 4) – (2 * 1) = 10

Characteristic Polynomial: λ² – 7λ + 10