Matrices

I.  Homework

II.  Definition of a Matrix

An m by n matrix is an array of numbers with m rows and n columns.

Example 1:
4 5
0 15
-9 3

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 yesterday.  

1)  interchanging two rows

2)  multiplying a row by a nonzero constant

3)  replacing a row with the row + a constant multiple of another row.

III.  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 we begin solving by performing row operations:

R1 <-> R2
1 1 1 2
2 -1 1 3
0 1 -1 -1

R2 - 2R1 -> R2
1 1 1 2
0 -3 -1 -1
0 1 -1 -1

R2 <-> R3
1 1 1 2
0 1 -1 -1
0 -3 -1 -1

R1  - R2 -> R1

R3 + 3R2 -> R3

1 0 2 3
0 1 -1 -1
0 0 -4 -4

R3 -> -1/4 R3
1 0 2 3
0 1 -1 -1
0 0 1 1

R1 - 2R3 -> R1

R2 + R3 -> R2
1 0 0 1
0 1 0 0
0 0 1 1

We can now put the matrix back in equation form:

x = 1, y = 0 and z = 1

If we have seen a bottom row that was of the form 0 0 0 a where a is a nonzero constant, then there are no solutions.  If a was 0 there would be infinitely many solutions.

IV.  Addition and Scalar Multiplication of Matrices

We can only add matrices that are of the same dimensions, that is if

A =
1 2
3 4

B =
2 3
4 1
5 9

and

C =
1 3
7 2

then only A + C makes sense.  We write

A + C=
1 + 1 2 + 3
3 + 7 2 + 4

=
2 5
10 6

For any matrix, we can multiply a matrix by a real number as in the following example (Same B as above):

5B =
10 15
20 5
25 45

We define the zero matrix to be the matrix with only zeros for entries.  For example the 2 by 2 zero matrix is
0 0
0 0

V.  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:

Let A =
3 4 2
0 1 -2

and B =
7 -3
-2 1
0 5

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)

=
13 5
-12 -9

Exercises:  Let  A =
1 2
3 4

B =
4 2 1
-2 0 0
1 6 -1

C =
1 0
2 1
4 5

D =
3 4 0
5 0 0

and E =
3 4 2
1 5 0
1 -1 2

Evaluate each one that makes sense:

1)  A + B    2)  4C     3)  AB     4)  CD      5)  DC     6)  B + E     7)  A3

II.  Applications of Matrices

We will work in groups to solve the following problems:

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