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

## 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:

- First, determine the trace of the matrix, which is the sum of the elements a
_{11}and a_{22}. - Next, calculate the determinant of the matrix, which is (a
_{11}* a_{22}) – (a_{12}* a_{21}). - Use the formula λ² – (trace(A))λ + det(A) to find the characteristic polynomial.
- 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:

a_{11} = 3, a_{12} = 2

a_{21} = 1, a_{22} = 4

Trace(A) = a_{11} + a_{22} = 3 + 4 = 7

Det(A) = (a_{11} * a_{22}) – (a_{12} * a_{21}) = (3 * 4) – (2 * 1) = 10

Characteristic Polynomial: λ² – 7λ + 10