Assignment 24

"If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy." -- (Alfred Renyi)

Math History Tidbit:

Many of the words we use to describe branches of mathematics are of Greek origin:

Arithmetic: From arithmetike, meaning "the art of counting."

Geometry: From geometrein, meaning "to measure the land."

Mathematics: From the combination of two words; one is manthanein, meaning "to learn." The other is mathema, meaning "science."

Herkimer's Corner

Why did Herkimer look for an ax on December 22?

Answer: Because there were only three chopping days left before Christmas.

Things Herky would like to know:

Why are the places called apartments when they are all stuck together?

Why do people press harder on the remote-control when they know the battery is dead?


Reading: Section 4.5, pages 230-232.

Written: Pages 233-234/ 23-26, 35-39. In each problem, write the linear system as a matrix equation. Then, use matrices on your graphics calculator to solve the system. Do this neatly. Don't be sloppy.

Items for reflection:

Mathematical word analysis:
CONGRUENT: From the Latin word congruens (to come together). In geometry, congruent figures are identical is size and shape, and become one figure if they "come together."

It's important to remember that matrix multiplication is not commutative. That is, if [A] and [B] are matrices, then

[A]x[B] is not necessarily equal to [B]x[A] . In fact, it's possible that the matrix product [A]x[B] could be a matrix, and that the matrix product [B]x[A] is not a matrix.

Matrix multiplication "logic" in solving systems is indicated below. This should make sense to you.

[A]x[X] = [B]

==> [A]-1x[A]x[X] = [A]-1x[B]

==> [X] = [A]-1x[B].

The key here is that if you are solving systems of equations involving 2 or more variables, [A]-1x[B] will be a matrix, but [B]x[A]-1 will not be a matrix. To repeat, matrix multiplication is not commutative.


Problem: Explain how to use matrix operations on the TI-83 calculator to solve the system

4x + y = 14
7x - 2y = 17

Solution (with communication):

Here is an outline of the calculator process:

MATRIX ---> EDIT ---> 1:[A]

MATRIX [A] 2 x 2
[4 1]
[7 -2]

MATRIX ---> EDIT ---> 2:[B]

MATRIX [B] 2 x 1

QUIT (2nd, then MODE)... gets you out of EDIT mode.

MATRIX ---> 1:[A] ---> X-1 ---> * ---> MATRIX ---> 2: [B] ---> ENTER

The calculator will display

The solution for the system is x = 3 and y = 2. The two lines intersect at the point (3,2).