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Definition of a Matrix
An m by n matrix is an array of numbers with
m rows and n columns.
Example 1:
is a 3 by 2 matrix.
Example 2:
Consider the system of equations
2x - y + 3z = 5
x +
4z = 3
5x - 7y + 3z = 7
Then the matrix
|
| 2 |
-1 |
3 |
5 |
| 1 |
0 |
4 |
3 |
| 5 |
-7 |
3 |
7 |
|
is called the augmented matrix associated to the system of equations.
Two matrices are called equal if all of their entries are the same.
Two matrices are called row equivalent is one can be transformed using a
sequence of the three operations that we discussed earlier.
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Interchanging two rows.
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Multiplying a row by a nonzero constant.
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Replacing a row with the row + a constant multiple of another row.
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Solving Linear Systems Using Matrices
We can solve a linear system by writing down its augmented matrix and performing
the row operations that we did last time.
Example:
Solve
2x - y + z = 3
x + y + z = 2
y - z = -1
Solution:
We write the associated augmented matrix:
|
| 2 |
-1 |
1 |
3 |
| 1 |
1 |
1 |
2 |
| 0 |
1 |
-1 |
-1 |
|
Now begin solving by performing row operations:
|
R1 <-> R2
|
R2 - 2R1 -> R2
|
R2 <-> R3
|
| 1 |
1 |
1 |
2 |
| 2 |
-1 |
1 |
3 |
| 0 |
1 |
-1 |
-1 |
|
| 1 |
1 |
1 |
2 |
| 0 |
3 |
-1 |
-1 |
| 0 |
1 |
-1 |
-1 |
|
| 1 |
1 |
1 |
2 |
| 0 |
1 |
-1 |
-1 |
| 0 |
3 |
-1 |
-1 |
|
R1 - R2 -> R1
R3 + 3R2 -> R3
|
R3 -> -1/4 R3
|
R1 - 2R3 -> R1
R2 + R3 -> R2 |
| 1 |
0 |
2 |
3 |
| 0 |
1 |
-1 |
-1 |
| 0 |
0 |
-4 |
-4 |
|
| 1 |
0 |
2 |
3 |
| 0 |
1 |
-1 |
-1 |
| 0 |
0 |
1 |
1 |
|
|
R1 - 2R3 -> R1
R2 + R3 -> R2
We can now put the matrix back in equation form:
x = 1, y = 0
and z = 1
Note: If we had seen a bottom row that was of the form 0 0 0 a where a is a nonzero
constant, then there would be no solution. If a had been 0 there would be infinitely
many solutions.
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Addition and Scalar Multiplication of Matrices
We can only add matrices that are of the same dimensions, that is if
then only A + C makes sense. We write
For any matrix, we can multiply a matrix by a real number as in the following
example (Same B as above):
We define the zero matrix to be the matrix with only zeros for entries.
For example, the 2 by 2 zero matrix is
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Multiplication of Matrices
To multiply matrices, unfortunately the definition is not the obvious one.
We can only multiply matrices where the number of columns of the first
matrix is the same as the number of rows of the second matrix. The
best way to learn how to multiply matrices is by example:
|
Then AB= |
| 3(7) + 4(-2) + 2(0) |
3(-3) + 4(1) + 2(5) |
| 0(7) + 1(-2) + -2(5) |
0(-3) + 1(1) + -2(5) |
|
= |
|
Exercises:
| Let A = |
|
,
B = |
|
, C = |
|
, D = |
|
, E = |
|
Evaluate each one that makes sense:
1) A + B 2) 4C 3) AB
4) CD 5) DC
6) B
+ E 7) A3
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Applications of Matrices
Application 1
A) Tables and chairs are made in the Mexico plant, the Brazil plant,
and the US plant. The matrix below represents the quantity made per
day.
|
A = |
Quantity
|
Mexico |
Brazil |
US |
| Tables |
15 |
10 |
50 |
| Chairs |
30 |
12 |
75 |
|
Labor and material cost for 1997 are represented in the following matrix.
|
B = |
|
Labor |
Material |
| Mexico |
15 |
20 |
| Brazil |
12 |
10 |
| US |
30 |
5 |
|
In 1997, the costs have increases to
| |
C = |
|
Labor |
Material |
| Mexico |
17 |
25 |
| Brazil |
15 |
15 |
| US |
45 |
10 |
|
Find the following and describe what they mean:
1) AB 2) C - B 3) AC
4) A(C - B) 5) 365AC
Application 2
Suppose that you have two jobs, each contribute to two different mutual funds
for retirement. The first fund pays 5% interest and the second pays
8% interest. Initially $5,000 is put into the funds and after one year
there will be $5,300. If the first fund got half of the money from
the first job and one third of the money from the second job, how much did
each job contribute?
Hint: Multiplication of matrices is the same as composition
of functions