Cayley Hamilton theorem. Hello friends, today it’s about Cayley Hamilton theorem in matrix analysis.

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### 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 is . Now here is the eigenvalue of the matrix . And is the unit matrix or the identity matrix.

Also, means the determinant of the matrix

Thus the matrix is

Therefore the characteristic equation of the matrix is

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 as

Since , then I can say

So this means

Now I’ll simplify it a bit more to get

Hence I can say that the characteristic equation of the matrix is

Next, I’ll show that the matrix will satisfy its characteristic equation. So that means I have to show that

(1)

###### Step 2

So I’ll start with . Now I can say

which means

Now I’ll evaluate it to get

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

Now will be

So this means

Now I’ll evaluate it to get

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

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 satisfies Cayley hamilton theorem.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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