Rank of a square matrix. Here it’s about the rank of a square matrix. Have a look!!

**Rank of a square matrix**

Suppose a matrix has 2 rows and 2 columns. Then it is a square matrix.

Also, any matrix with 3 rows and 3 columns will also be a square matrix.

But if the matrix has 2 rows and 3 columns, it will not be a square matrix anymore. Then it’s a rectangular matrix. This means a matrix with an equal number of rows and columns is a square matrix.

Here I’m talking about the rank of such a matrix.

Now the rank of a matrix is the highest order of the matrix with a non-zero determinant.

In another way, I can also say that the rank is the number linearly independent rows of a matrix. So here I’ll solve some examples of the rank of a square matrix. I hope that’ll give a better idea.

Later I’ll use the rank of a square matrix to find the nature of solutions of a set of equations without solving them.

See also: Rank of a rectangular matrix

**Solved examples of the rank of a square matrix**

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

So here are the examples.

**Example 1**

According to Kreyszig (2005) “Find the rank of

”

**Solution**

Here the given matrix is

Let me give it a name, say, .

So the matrix is

I can see the matrix has 3 rows and 3 columns. So the matrix is a square matrix.

In order to get the rank of this matrix, first of all, I’ll try to reduce it to an upper triangular matrix.

**Step 1**

First of all, I’ll subtract row 2 from row 3.

In mathematical term, I’ll write like:

Row 3 – Row 2.

Thus the equivalent matrix will be

Then I interchange row 1 and row 3 to get the equivalent matrix as

Now I’ll subtract 3 times row 1 from row 2.

Therefore Row 2 – 3 Row 1 gives

Again I’ll interchange rows. Now it’ll be row 2 and row 3. Thus the equivalent matrix is

Next, I’ll add 5 times row 2 to row 3.

Thus Row 3 + 5 Row 2 gives

Now here I have to stop. This is because I can’t reduce it further.

My next step is to find out the value of the determinant.

**Step 2**

Thus the determinant of the matrix is

Therefore the rank of the matrix cannot be 3.

So now I choose any submatrix of , say

Next, I’ll evaluate the determinant of . So it will be

Thus I can say that the rank of the matrix is 2.

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

Now I’ll go to the next example.

**Example 2**

According to Kreyszig (2005) “Find the rank of

”

**Solution**

Here the given matrix is

Let me give it a name, say, .

So the matrix is

I can see the matrix has 4 rows and 4 columns. So the matrix is a square matrix.

In order to get the rank of this matrix, first of all, I’ll try to reduce it to an upper triangular matrix.

**Step 1**

First of all, I’ll subtract twice row 1 from row 2.

Simultaneously, I’ll also subtract thrice row 1 from row 3.

I’ll also subtract 4 times row 1 from row 4.

In mathematical terms, I’ll write like:

Row 2 – 2 Row 1, Row 3 – Row 1, Row 4 – 4 Row 1.

Thus the equivalent matrix will be

Now I’ll subtract 2 times row 2 from row 3.

Simultaneously, I’ll also subtract 7 times row 2 from row 4.

Therefore Row 3 – 2 Row 2, Row 4 – 7 Row 2 give

Next, I’ll add row 4 with row 1.

Therefore Row 4 + Row 1 gives

Here I have to stop. This is because I can’t reduce it further.

My next step is to find out the value of the determinant.

**Step 2**

Thus the determinant of the matrix is

Therefore I can say that the rank of the matrix is 4.

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

Now I’ll go to the next example.

**Example 3**

According to Kreyszig (2005) “Find the rank of

”

**Solution**

Here the given matrix is

Let me give it a name, say, .

So the matrix is

I can see the matrix has 4 rows and 4 columns. So the matrix is a square matrix.

In order to get the rank of this matrix, first of all, I’ll try to reduce it to an upper triangular matrix.

**Step 1**

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

Simultaneously, I’ll also add row 4 to row 2.

I’ll also subtract 4 times row 1 from row 4.

In mathematical terms, I’ll write like:

Row 3 – 3 Row 1, Row 4 + Row 2.

Thus the equivalent matrix will be

Now I’ll add 4 times row 3 to row 4.

Therefore Row 4 + 4 Row 3 gives

Here I have to stop. This is because I can’t reduce it further.

My next step is to find out the value of the determinant.

**Step 2**

Thus the determinant of the matrix is

Therefore the rank of the matrix cannot be 4.

So now I choose any 33 submatrix of , say

Next, I’ll evaluate the determinant as

Thus I can say that the rank of the matrix is 3.

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

Dear friends, this is the end of 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!!

MohTarek says

The first example isnt correct i hope u change it …. A22 after Row 2 – 3 Row 3 should be ” -15 ” not ” 15 “

Dr. Aspriha Peters says

Hi,

Yes, of course, you are absolutely right. Thanks a ton for pointing it out. I’ve already changed it. Once again, thank you very much.