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

**Rank of a rectangular 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 unequal number of rows and columns is a rectangular 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 of linearly independent rows of a matrix. So here I’ll solve some examples of the rank of a rectangular matrix. I hope that’ll give a better idea.

Read also: Rank of a square matrix

**Examples on the rank of a rectangular 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 on the rank of a rectangular matrix.

**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 4 rows and 3 columns. So the matrix is a rectangular 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 half of row 1 from row 3.

Simultaneously, I’ll also subtract half of row 2 from row 4.

In mathematical term, I’ll write like:

Row 3 – 1/2 Row 1 and Row 4 – 1/2 Row 2.

Thus the equivalent matrix will be

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

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 of any submatrix of the matrix .

**Step 2**

So, I’ll start with a 3 × 3 submatrix of .

First of all, I’ll choose a 3 × 3 submatrix, say

Let me give this matrix a name, say .

Thus the determinant of the matrix is

Now I’ll expand and evaluate the determinant to get

Then I’ll simplify it to get

So this means

Therefore I can say that the rank of the matrix is **not** 3.

Next, I’ll choose a 2 × 2 submatrix of , say

Let me give this matrix a name, say .

Thus the determinant of the matrix is

Again I’ll expand and evaluate the determinant to get the value of as

Next, I’ll simplify it to get

Therefore 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 3 rows and 4 columns. So the matrix is a rectangular 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 5 times row 2 from row 3.

In mathematical terms, I’ll write like:

Row 3 – 5 Row 2.

Thus the equivalent matrix will be

Now I’ll interchange row 1 and row 2.

Therefore the equivalent matrix will be

Next, I’ll subtract 15/2 times row 2 from row 3.

Therefore Row 3 – 15/2 Row 2 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 of any submatrix of the matrix .

**Step 2**

So, I’ll start with a 3 × 3 submatrix of .

First of all, I’ll choose a 3 × 3 submatrix, say

Let me give this matrix a name, say .

Thus the determinant of the matrix is

Now I’ll expand the determinant as

Then I’ll evaluate the determinant as

So this means that the determinant of the matrix is . And this is not equal to .

Therefore 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 on the rank of a rectangular 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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