**Statement**

Cayeley-Hamilton theorem-

According to Cayeley-Hamilton theorem, every square matrix satisfies its characteristic equation, that means for every square matrix of order n,

Then the matrix equation-

Is satisfied by X = A

That means

**Example-1: Find the characteristic equation of the matrix given below and Verify cayley-Hamlton theorem-**

Sol. Characteristic equation of the matrix, we can be find as follows-

Which is-

According to cayley-Hamilton theorem,

Now we will verify equation (1),

Put the required values in equation (1), we get

Hence the cayley-Hamilton theorem is verified.

**Example: Find the characteristic equation of the the matrix A and verify Cayley-Hamilton theorem as well.**

Sol. Characteristic equation will be-

Which gives,

Or

According to cayley-Hamilton theorem,

In order to verify cayley-Hamilton theorem , we will find the values of and

So that,

Now

Put these values in equation(1), we get

Hence the cayley-hamilton theorem is verified.

**Inverse of a matrix by Cayley-Hamilton theorem**

We can find the inverse of any matrix by multiplying the characteristic equation with A-1.

For example,

suppose we have a characteristic equation A^{3}-5A^{2}+7A-3I=0 then multiply this by A-1 , then it becomes

A^{2}-5A+7I-3A^{-1}=0

Then we can find A^{-1} by solving the above equation.

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