Page 52 - Matrix Analysis & Applied Linear Algebra
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2.1 Row Echelon Form and Rank 45
is extremely important in the development of our subject.
Rank of a Matrix
Suppose A m×n is reduced by row operations to an echelon form E.
The rank of A is defined to be the number
rank (A) = number of pivots
= number of nonzero rows in E
= number of basic columns in A,
where the basic columns of A are defined to be those columns in A
that contain the pivotal positions.
Example 2.1.2
Problem: Determine the rank, and identify the basic columns in
1211
A = 2422 .
3634
Solution: Reduce A to row echelon form as shown below:
1
211 21 1 21 1
1
1
A = 2 422 −→ 0 0 0 −→ 0 0 0 = E.
0
1
3 634 0 0 0 1 0 0 0 0
Consequently, rank (A)=2. The pivotal positions lie in the first and fourth
columns so that the basic columns of A are A ∗1 and A ∗4 . That is,
1 1
2
Basic Columns = , .
2
3 4
Pay particular attention to the fact that the basic columns are extracted from
A and not from the row echelon form E .
