Cayley-Hamilton Theorem

For a square matrix , identity matrix , and zero matrix , the following holds:

This is called the Cayley-Hamilton Theorem.

[Proof] Given,

Thus,

[Note] For , the Cayley-Hamilton theorem gives . However, matrix is not unique in satisfying the equation . For example, also satisfies .

Example. Given the matrix satisfies the following equation.

Find the value offor the constantsand.

 Solution

Since ( is a real number), by the Cayley-Hamilton theorem,

Therefore,