The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. This means that if A is an n × n matrix with characteristic polynomial p(t), then p(A) = 0, where 0 is the zero matrix of size n × n. This provides a way to find the characteristic polynomial of a matrix without directly computing the determinant.

Use the Cayley-Hamilton Theorem to Find the Characteristic Polynomial of a Matrix

Steps

To use the Cayley-Hamilton theorem to find the characteristic polynomial of a matrix, follow these steps:

Find the characteristic equation of the matrix A by computing det(A – tI), where I is the n × n identity matrix.

Substitute the matrix A for the variable t in the characteristic equation to obtain p(A).

Simplify p(A) by using the properties of matrix multiplication and addition.

Set p(A) = 0 to obtain the characteristic polynomial of A.

Example

Consider the matrix A = \begin{bmatrix}1 & 2 \ 2 & 1\end{bmatrix}. To find its characteristic polynomial using the Cayley-Hamilton theorem, we first compute the characteristic equation:

det(A – tI) = \begin{vmatrix}1 – t & 2 \ 2 & 1 – t\end{vmatrix} = (1 – t)^2 – 4 = t^2 – 2t – 3

Next, we substitute A for t in the characteristic equation to obtain:

p(A) = A^2 – 2A – 3I

Using the matrix multiplication properties, we can simplify this to:

p(A) = \begin{bmatrix}1 & 2 \ 2 & 1\end{bmatrix} \begin{bmatrix}1 & 2 \ 2 & 1\end{bmatrix} – 2 \begin{bmatrix}1 & 2 \ 2 & 1\end{bmatrix} – 3 \begin{bmatrix}1 & 0 \ 0 & 1\end{bmatrix} = \begin{bmatrix}-3 & 2 \ 2 & -3\end{bmatrix}

Finally, we set p(A) = 0 to obtain the characteristic polynomial of A:

p(t) = t^2 – 2t – 3

Therefore, the characteristic polynomial of A is t^2 – 2t – 3.

Conclusion

The Cayley-Hamilton theorem provides a useful method for finding the characteristic polynomial of a matrix without directly computing the determinant. By substituting the matrix for the variable in the characteristic equation and simplifying using the properties of matrix multiplication and addition, we can obtain the characteristic polynomial.