Page 232 - Advanced engineering mathematics

212 CHAPTER 7 Matrices and Linear Systems

EXAMPLE 7.17
Let
⎛ ⎞
0100 3 0 6
0010 −2 1 5
⎜ ⎟
A = ⎜ ⎟ .
⎝ 0001 2 0 −4 ⎠
0000 0 0 0
Since this is a reduced matrix with three nonzero rows, rank(A) = 3.

COROLLARY 7.2
Let A be an n × n matrix of real numbers. Then
rank(A) = n if and only if A R = I n .

This says that the rank of a square matrix equals the number of rows exactly when the
reduced form is the identity matrix.
Proof First, we know that
rank(A) = number of nonzero rows of A R .
If A R = I n , then A R has n nonzero rows and this matrix has rank n, hence A also has rank n.
Conversely, if A has rank n, then so does A R , so this reduced matrix is an n × n matrix with
1 down the main diagonal and all other elements (above and below leading entries) zero. Then
A R = I n .

The MAPLE command rank(A) will return the rank of A.

SECTION 7.4 PROBLEMS

In each of Problems 1 through 14, find the reduced form of 6. 1 3 0
the matrix and use this to determine the rank of the matrix. 0 0 1
Also find a basis for the row space of the matrix and a basis ⎛ 2 2 1 ⎞
for the column space.
⎜ 1 −1 3 ⎟
7. ⎜ ⎟
⎝ 0 0 1 ⎠
4 0 7
−4 1 3
1.
2 2 0 ⎛ ⎞
0 −1 0
8. ⎝ 0 0
⎛ ⎞ −1 ⎠
1 −1 4
0 0 2
2. ⎝ 0 1 3 ⎠
2 −1 11 ⎛ 0 4 3 ⎞
9. ⎝ 0 1
⎛ ⎞ 0 ⎠
−3 1 2 2 2
3. ⎝ 2 2 ⎠
⎛ ⎞
4 −3 1 0 0
2 0 0
⎜ ⎟
10.
⎝ 1
⎛ ⎞ ⎜ ⎟
6 0 0 1 1 0 −1 ⎠
4. ⎝ 12 0 0 2 2 ⎠ 3 0 0
1 −1 0 0 0 ⎛ ⎞
−3 2 2
11. ⎝ 1 0
8 −4 3 2 5 ⎠
5. 0 0 2
1 −1 1 0
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October 14, 2010 14:23 THM/NEIL Page-212 27410_07_ch07_p187-246