What is Cayeley-Hamilton Theorem?

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  A³-5A²+7A-3I=0  then multiply this by A-1 , then it becomes

A²-5A+7I-3A^-1=0

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

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