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Introduction to Matrices

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides an introduction which develops concepts related to matrices.

The following matrix, stolen from a rusted lockbox in the back of a large, dark lecture hall in a school called Hogwart’s, is the gradebook for Professor Severus Snape’s class in potions.

Table 1
  Poison Cure Love philter Invulnerability
Granger, H 100 105 99 100
Longbottom, N 80 90 85 85
Malfoy, D 95 90 0 85
Potter, H 70 75 70 75
Weasley, R 85 90 95 90

When I say this is a “matrix” I’m referring to the numbers in boxes. The labels (such as “Granger, H” or “Poison”) are labels that help you understand the numbers in the matrix, but they are not the matrix itself.

Each student is designated by a row. A row is a horizontal list of numbers.

Exercise 1

Below, copy the row that represents all the grades for “Malfoy, D.”

Each assignment is designated by a column, which is a vertical list of numbers. (This is easy to remember if you picture columns in Greek architecture, which are big and tall and…well, you know…vertical.)

Exercise 2

Below, copy the column that represents all the grades on the “Love philter” assignment.

I know what you’re thinking, this is so easy it seems pointless. Well, it’s going to stay easy until tomorrow. So bear with me.

The dimensions of a matrix are just the number of rows, and the number of columns…in that order. So a “10 × 20” matrix means 10 rows and 20 columns.

Exercise 3

What are the dimensions of Dr. Snape’s gradebook matrix?

For two matrices to be equal, they must be exactly the same in every way: same dimensions, and every cell the same. If everything is not precisely the same, the two matrices are not equal.

Exercise 4

What must xx and yy be, in order to make the following matrix equal to Dr. Snape’s gradebook matrix?

Table 2
100 105 99 100
80 x+yx+y 85 85
95 90 0 85
70 75 x-yx-y 75
85 90 95 90

Finally, it is possible to add or subtract matrices. But you can only do this when the matrices have the same dimensions!!! If two matrices do not have exactly the same dimensions, you cannot add or subtract them. If they do have the same dimensions, you add and subtract them just by adding or subtracting each individual cell.

Exercise 5

As an example: Dr. Snape has decided that his grades are too high, and he needs to curve them downward. So he plans to subtract the following grade-curving matrix from his original grade matrix.

Table 3
5 0 10 0
5 0 10 0
5 0 10 0
10 5 15 5
5 0 10 0

Write down the new grade matrix.

Exercise 6

In the grade-curving matrix, all rows except the fourth one are identical. What is the effect of the different fourth row on the final grades?

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