Page 246 - Advanced engineering mathematics

226 CHAPTER 7 Matrices and Linear Systems

The first three columns tell us that A has a rank of 3, so the associated homogeneous system
.
.
has only the trivial solution. Since the rank of [A R .C] is also 3, the system has a solution. This
solution is unique because A R = I 3 .
.
.
From the fourth column of [A.B] R , we read the unique solution
⎛ ⎞
−86/31
.
X = −191/155 ⎠
⎝
−11/155

SECTION 7.6 PROBLEMS

In each of Problems 1 through 14, find the general solution 8. 2x 1 − 3x 3 = 1
of the system or show that the system is inconsistent. Write x 1 − x 2 + x 3 = 1
the solution in matrix form. 2x 1 − 4x 2 + x 3 = 2

1. 3x 1 − 2x 2 + x 3 = 6 9. 14x 3 − 3x 5 + x 7 = 2
x 1 + 10x 2 − x 3 = 2
x 1 + x 2 + x 3 − x 4 + x 6 =−4
−3x 1 − 2x 2 + x 3 = 0
2. 4x 1 − 2x 2 + 3x 3 + 10x 4 = 1 10. 3x 1 − 2x 2 =−1
x 1 − 3x 4 = 8 4x 1 + 3x 2 = 4
2x 1 − 3x 2 + x 4 = 16
11. 7x 1 − 3x 2 + 4x 3 == −7
3. 2x 1 − 3x 2 + x 4 − x 6 = 0
3x 1 − 2x 3 + x 5 = 1 2x 1 + x 2 − x 3 + 4x 4 = 6
x 2 − x 4 + 6x 6 = 3 x 2 − 3x 4 =−5
4. 2x 1 − 3x 2 = 1 12. −4x 1 + 5x 2 − 6x 3 = 2
−x 1 + 3x 2 = 0 2x 1 − 6x 2 + x 3 =−5
x 1 − 4x 2 = 3 −6x 1 + 16x 2 − 11x 3 = 1
5. 3x 2 − 4x 4 = 10
x 1 − 3x 2 + 4x 3 − x 6 = 8 13. 4x 1 − x 2 + 4x 3 = 1
x 2 + x 3 − 6x 4 + x 6 =−9 x 1 + x 2 − 5x 3 = 0
x 1 − x 2 + x 6 = 0 −2x 1 + x 2 + 7x 3 = 4
6. 2x 1 − 3x 2 + x 4 = 1
3x 2 + x 3 − x 4 = 0 14. −6x 1 + 2x 2 − x 3 + x 4 = 0
x 1 + 4x 2 − x 4 =−5
2x 1 − 3x 2 + 10x 3 = 0
x 1 + x 2 + x 3 − 7x 4 = 0
7. 8x 2 − 4x 3 + 10x 6 = 1
x 3 + x 5 − x 6 = 2 15. Show that the system AX=B is consistent if and only
x 4 − 3x 5 + 2x 6 = 0 if B is in the column space of A.

7.7 Matrix Inverses

Let A be an n × n matrix. An n × n matrix B is an inverse of A if
AB = BA = I n .

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:23 THM/NEIL Page-226 27410_07_ch07_p187-246