section 7.1 matrices and systems of linear equations
TRANSCRIPT
Section 7.1
Matrices and Systems of Linear Equations
Matrices
A matrix is a rectangular array of numbers written within brackets
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Order of a Matrix
Order of a Matrix: Row x Column
3 rows, 2 columns 3 x 2
6448
4324
3233
81102
4 rows, 4 columns 4 x 4
652 1 row, 3 columns 1 x 3
89
34
62
Augmented Matrices
Matrices can be used as shorthand for systems of equations. When done so, they are called augmented matrices.
72
143
yx
yx
7
1
21
43
Each row is an equation
Vertical line represents the equal sign
First column is coefficients on the x
Second column is coefficients on the y
Constants to the right of the vertical line
Any variable not in the equation has an implied coefficient of 0
Write the system as an augmented matrix
0342
723
zx
zyx
3
7
402
231
1
2
7
yz
zyx
zxy
1
2
7
110
111
111
Row Operations (Solving Systems)
Interchange any 2 rows
Multiply a row by a nonzero constant
Add a multiple of 1 row to another
ji RR
ii RcR
jji RRcR
Perform the row operation
1
3
20
1221 RR
3
1
12
20
Perform the row operation
4
1
2
516
423
175222 RR
4
2
516
175
Perform the row operation
2
3
3
124
210
201131 2 RRR
2
3
124
210
Perform the row operation
1
3
2
310
213
101
121 3 RRR
1
3
310
213
Row-Echelon Form of a Matrix
Rows consisting entirely of 0’s are at the bottom of the matrix
For each row that does not consist entirely of 0’s, the first (leftmost) nonzero entry is 1 (called the leading 1)
The leading 1 in each row must have all zeros underneath it.
0000
5100
1210
6531
Determine whether the matrices are in Row-Echelon Form
7
6
5
100
410
221
7
1
5
401
510
621
0
3
5
000
1510
26121
7
2
10
100
640
871
Yes No
Yes No
Rewrite the Matrix in Row Echelon Form
3
2
11
23
21 RR
2
3
23
11
11
3
10
11
221 RR
11
3
10
11
2213 RRR
Solve the system using Gaussian Elimination
63
82
yx
yx
6
8
31
12Step 1: Write as an augmented matrix
8
6
12
31
Step 2: Use row operations to write in row-echelon form.
21 RR
Need a 0 below the leading 1 in row 1
20
6
50
31
2212 RRR
20
6
50
31
…continued
Need a leading 1 in row 2 (turn the -5 into a 1)
225
1RR
4
6
10
31
Step 3: Write the augmented matrix as a system of equations.
4
63
y
yx
Step 4: Back substitute to find all other variables.
643 x
612 x6x
6x 4y
Solve the system
3322
43
2
zyx
zyx
zyx
3
4
2
322
113
111
7
2
2
540
220
111
2213 RRR
and
3312 RRR
7
2
2
540
220
111
7
1
2
540
110
111222
1RR
3
1
2
100
110
1113324 RRR
3
1
2
z
zy
zyx
13 y2y
232 x21 x1x
1x 2y 3z
Infinitely Many and No Solutions
0
4
2
000
610
321 Row 3 equation would say:
0x + 0y + 0z = 0
0 = 0
Infinitely Many Solutions on a line
4
4
2
000
610
321 Row 3 equation would say:
0x + 0y + 0z = 4
0 = 4
No Solutions