Gauss-Jordan method to find out the inverse of a matrix. Hello friends, today it’s about the Gauss-Jordan method to find out the inverse of a matrix.

**Gauss-Jordan method to find out the inverse of a matrix**

Let’s say I have a matrix

And I want to find out the inverse of this matrix.

Now, to get the inverse of the matrix , I will follow a few steps.

- First of all, I will find out the determinant of the matrix.
- Next, I will determine the cofactor of each element of the matrix.
- Then I’ll write to them in a matrix form.
- In the end, I will find out the transpose of the new matrix.

And then only I can get the inverse of the matrix by using another formula.

If interested, you can read all the details procedure in my post on the inverse of a matrix.

But today I’ll use the Gauss-Jordan method to find out the inverse of a matrix. And you will see that it’s quite a straight forward thing.

**Gauss-Jordan method**

Now in the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side. So the resultant matrix will look like

And my aim is to bring the unit matrix on the left-hand side. And for that, I have to use row operations on this matrix.

As per the Gauss-Jordan method, the matrix on the right-hand side will be the inverse of the matrix .

Now I’ll give some examples of how to use the Gauss-Jordan method to find out the inverse of a matrix.

** Solved examples of Gauss-Jordan method to find out the inverse of a matrix**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

Here comes my first example.

** Example 1**

According to Kreyszig (2005)*, “Find the inverse by Gauss-Jordan.

”

**Solution**

Now here the given matrix is

First of all, I’ll give it a name, say . Therefore the given matrix is

Now I’ll use the Gauss-Jordan method to find out the inverse of the matrix .

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. As per the Gauss-Jordan method, the matrix on the right-hand side will be the inverse of the matrix .

**Step 1**

First of all, I’ll add times row 1 to row 2. Simultaneously, I’ll subtract twice row 1 from row 3.

In mathematical form, I’ll write like:

So the resultant matrix is

As I can see, I have already got the identity matrix component at the second row.

Now I’ll try to bring the other two as well.

**Step 2**

Next, I’ll multiply the row 3 of the matrix with .

So in mathematical form, it will be:

Then the resultant matrix is

Now I’ll interchange row 1 and 3 to get the resultant matrix as

Now I have got the identity matrix component at the first row as well.

Next, I’ll do that with the third row too.

**Step 3**

Now I’ll subtract 3 times row 1 from row 3.

So in mathematical form, it will be:

Then the resultant matrix will be

Next, I’ll add row 2 to row 3. So in mathematical form, it will be:

Now I get the resultant matrix as

As I can see, the unit matrix is on the left-hand side of the matrix .

So this means the right-hand side matrix is the inverse of the matrix .

Thus the inverse of the matrix is

Hence I can conclude that this is the answer to this example.

Now comes my other example.

** Example 2**

According to Kreyszig (2005)*, “Find the inverse by Gauss-Jordan.

”

**Solution**

Now here the given matrix is

First of all, I’ll give it a name, say . Therefore the given matrix is

Now I’ll use the Gauss-Jordan method to find out the inverse of the matrix .

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. As per the Gauss-Jordan method, the matrix on the right-hand side will be the inverse of the matrix .

**Step 1**

First of all, I’ll subtract twice row 1 from row 3.

In mathematical form, I’ll write like:

So the resultant matrix is

As I can see the last row of the left-hand side matrix has all zero, I can’t calculate further.

Hence the conclusion is that this matrix doesn’t have an inverse. And this is the answer to this example.

And now comes my last example.

** Example 3**

According to Kreyszig (2005)*, “Find the inverse by Gauss-Jordan.

”

**Solution**

Now here the given matrix is

First of all, I’ll give it a name, say . Therefore the given matrix is

Now I’ll use the Gauss-Jordan method to find out the inverse of the matrix .

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. As per the Gauss-Jordan method, the matrix on the right-hand side will be the inverse of the matrix .

**Step 1**

First of all, I’ll add twice row 1 to row 2.

In mathematical form, I’ll write like:

So the resultant matrix is

Now I’ll interchange row 2 and 3 to get the resultant matrix as

As I can see, I have already got the identity matrix component in the third row.

Now I’ll try to bring the other two as well.

**Step 2**

Next, I’ll add times row 3 to row 1. Simultaneously, I’ll subtract twice row 3 from row 2.

In mathematical form, I’ll write like:

Then the resultant matrix is

Now I’ll multiply the row 2 of the matrix with .

So in mathematical form, it will be:

Thus the resultant matrix is

Now I have got the identity matrix component at the second row as well.

Next, I’ll do that with the first row too.

**Step 3**

Now I’ll subtract twice row 2 from row 1.

In mathematical form, I’ll write like:

So the resultant matrix is

As I can see, the unit matrix is on the left-hand side of the matrix .

So this means the right-hand side matrix is the inverse of the matrix .

Thus the inverse of the matrix is

Hence I can conclude that this is the answer to this example.

Dear friends, this is the end of today’s post on the Gauss-Jordan method to find out the inverse of a matrix. 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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