156 CHAPTER 8. RANK OF A MATRIX

Example 8.3.5 Find the rank of the matrix1 2 1 3 0−4 3 2 1 23 2 1 6 54 −3 −2 1 7

 .

From the above definition, all you have to do is find the row reduced echelon form andthen count up the number of nonzero rows. But the row reduced echelon form of thismatrix is 

1 0 0 0 − 174

0 1 0 0 10 0 1 0 − 45

40 0 0 1 9

2

and so the rank of this matrix is 4.

Find the rank of the matrix 1 2 1 3 0−4 3 2 1 23 2 1 6 50 7 4 10 7

The row reduced echelon form is

1 0 0 32

52

0 1 0 −4 −170 0 1 19

2632

0 0 0 0 0

and so this time the rank is 3.

8.3.2 Finding The Row And Column Space Of A MatrixThe row reduced echelon form also can be used to obtain an efficient description of the rowand column space of a matrix. Of course you can get the column space by simply sayingthat it equals the span of all the columns but often you can get the column space as the spanof fewer columns than this. This is what we mean by an “efficient description”. This isillustrated in the next example.

Example 8.3.6 Find the rank of the following matrix and describe the column and rowspaces efficiently.  1 2 1 3 2

1 3 6 0 23 7 8 6 6

 (8.1)