Fundamentals of Matrix Algebra, 2011a

1.2 Using Matrices To Solve Systems of Linear Equaons
Noce how the second equaon shows that x 2 = 1. All that remains to do is to
solve for x 1 .
Replace equaon 1 with the sum
of (−1) mes equaon 2 plus
equaon 1
x 1 = −1
x 2 = 1
x 3 = 0
Replace row 1 with the sum of
(−1) mes row 2 plus row 1
(−R 2 + R 1 → R 1 )
⎡
⎣ 1 0 0 −1
0 1 0 1
0 0 1 0
⎤
⎦
Obviously the equaons on the le tell us that x 1 = −1, x 2 = 1 and x 3 = 0, and
noce how the matrix on the right tells us the same informaon. .
Exercises 1.2
In Exercises 1 – 4, convert the given system of
linear equaons into an augmented matrix.
1.
2.
3.
4.
3x + 4y + 5z = 7
−x + y − 3z = 1
2x − 2y + 3z = 5
2x + 5y − 6z = 2
9x − 8z = 10
−2x + 4y + z = −7
x 1 + 3x 2 − 4x 3 + 5x 4 = 17
−x 1 + 4x 3 + 8x 4 = 1
2x 1 + 3x 2 + 4x 3 + 5x 4 = 6
3x 1 − 2x 2 = 4
2x 1 = 3
−x 1 + 9x 2 = 8
5x 1 − 7x 2 = 13
In Exercises 5 – 9, convert the given augmented
matrix into a system of linear equa-
ons. Use the variables x 1 , x 2 , etc.
5.
[ 1 2
] 3
−1 3 9
6.
[ −3 4 7
]
0 1 −2
7.
[ 1 1 −1 −1
] 2
2 1 3 5 7
⎡
1 0 0 0 2
⎤
8.
⎢ 0 1 0 0 −1
⎥
⎣ 0 0 1 0 5 ⎦
0 0 0 1 3
9.
[ 1 0 1 0 7
] 2
0 1 3 2 0 5
In Exercises 10 – 15, perform the given row
operaons on A, where
⎡
2 −1 7
⎤
A = ⎣ 0 4 −2 ⎦ .
5 0 3
10. −1R 1 → R 1
11. R 2 ↔ R 3
12. R 1 + R 2 → R 2
13. 2R 2 + R 3 → R 3
1
14. R 2 2 → R 2
15. − 5 R1 + R3 → R3
2
A matrix A is given below. In Exercises 16 –
20, a matrix B is given. Give the row opera-
on that transforms A into B.
⎡ ⎤
1 1 1
A = ⎣ 1 0 1 ⎦
1 2 3
⎡
1 1
⎤
1
16. B = ⎣ 2 0 2 ⎦
1 2 3
⎡ ⎤
1 1 1
17. B = ⎣ 2 1 2 ⎦
1 2 3
⎡ ⎤
3 5 7
18. B = ⎣ 1 0 1 ⎦
1 2 3
11