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 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.

##### 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

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

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

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 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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