Recall the following definition of rank of a matrix.
Definition 4.2.1 A submatrix of a matrix A is the rectangular array of numbers obtained by deleting some rows and columns of A. Let A be an m × n matrix. The determinant rank of the matrix equals r where r is the largest number such that some r × r submatrix of A has a non zero determinant. The row rank is defined to be the dimension of the span of the rows. The column rank is defined to be the dimension of the span of the columns. The rank of A is denoted as rank
The following theorem is proved in the section on the theory of the determinant and is restated here for convenience.
Theorem 4.2.2 Let A be an m × n matrix. Then the row rank, column rank and determinant rank are all the same.
So how do you find the rank? It turns out that row operations are the key to the practical computation of the rank of a matrix.
In rough terms, the following lemma states that linear relationships between columns in a matrix are preserved by row operations.
Lemma 4.2.3 Let B and A be two m × n matrices and suppose B results from a row operation applied to A. Then the k^{th} column of B is a linear combination of the i_{1},
Proof: Let A equal the following matrix in which the a_{k} are the columns

and let B equal the following matrix in which the columns are given by the b_{k}

Then by Theorem 4.1.6 on Page 336 b_{k} = Ea_{k} where E is an elementary matrix. Suppose then that one of the columns of A is a linear combination of some other columns of A. Say

Then multiplying by E,

Corollary 4.2.4 Let A and B be two m×n matrices such that B is obtained by applying a row operation to A. Then the two matrices have the same rank.
Proof: Lemma 4.2.3 says the linear relationships are the same between the columns of A and those of B. Therefore, the column rank of the two matrices is the same. ■
This suggests that to find the rank of a matrix, one should do row operations until a matrix is obtained in which its rank is obvious.
Example 4.2.5 Find the rank of the following matrix and identify columns whose linear combinations yield all the other columns.
 (4.1) 
Take

By the above corollary, this matrix has the same rank as the first matrix. Now take

At this point it is clear the rank is 2. This is because every column is in the span of the first two and these first two columns are linearly independent.
Example 4.2.6 Find the rank of the following matrix and identify columns whose linear combinations yield all the other columns.
 (4.2) 
Take

Now multiply the second row by 1∕5 and add 5 times it to the last row.

Add
 (4.3) 
It is now clear the rank of this matrix is 2 because the first and third columns form a basis for the column space.
The matrix 4.3 is the row reduced echelon form for the matrix 4.2.