Cayley Hamilton theorem. Hello friends, today it’s about Cayley Hamilton theorem in matrix analysis. Have a look!!

### Cayley Hamilton theorem

Two noted mathematicians, Arthur Cayley, and William Rowan Hamilton discovered this theorem.

It says that every square matrix satisfies its own characteristic equation.

Now, this theorem has several applications in engineering. For example,

- it has an important role in computer programming and coding.
- it helps to determine the equations that illustrate the nature of materials in rheology.

Now I’ll give an example on this theorem.

#### A solved example in Cayley Hamilton theorem

So here is the example.

##### Example

Show that the matrix

satisfies Cayley Hamilton theorem.

##### Solution

Now here the given matrix is

First of all, I’ll give it a name, say .

So I can say,

As a first step, I’ll get the characteristic equation of the matrix

###### Step 1

So the characteristic equation of any matrix

Also,

Thus the matrix

Therefore the characteristic equation of the matrix

Next I will simplify the left-hand side of the equation.

So that means I’ll expand the determinant.

Therefore it will be

Then I’ll evaluate it to find the value of

Since

So this means

Now I’ll simplify it a bit more to get

Hence I can say that the characteristic equation of the matrix

Next, I’ll show that the matrix

(1)

###### Step 2

So I’ll start with

which means

Now I’ll evaluate it to get

Next, I’ll simplify it to get the value of

Now

So this means

Now I’ll evaluate it to get

Next, I’ll simplify it to get the value of

Now comes the next step.

###### Step 3

Thus from equation (1), I can say that

So this gives

Now this means

Hence I can conclude that the matrix

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