MatricesMatrix
Radha, Fauzia and Simran are getting ready for their exams. Radha has 15 notebooks and 6 pens, Fauzia has 10 notebooks and 2 pens, and Simran has 13 notebooks and 5 pens. A table keeps all of this tidy:
| Notebooks | Pens | |
|---|---|---|
| Radha | 15 | 6 |
| Fauzia | 10 | 2 |
| Simran | 13 | 5 |
Once everyone agrees that the rows are the three friends, in this order, and the columns are notebooks and pens, the labels can go. The bare numbers, each in its place, still say everything:
The number 10, in the second row and the first column, stands for
We could just as well have put the friends in the columns and the items in the rows: . Now the first
Definition 1 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or entries of the matrix.
We name matrices with capital letters. Here are three more:
The horizontal lines of
Tip
Rows run across, like the rows of seats in a cinema hall. Columns stand up, like the pillars of a building. Almost every rule in this chapter depends on which is which.
Order of a Matrix
A matrix with rows and columns has
Every element has an address. The element in row and column is written and called the th element:
or, in short, , where and . Row is , and column is .
Element hunt. Tap the three elements named in the missions. The row you are in lights up blue and the column green.
Tip
Rows always come first: in an order (a matrix has 3 rows and 2 columns) and in an address ( is in row 2, column 3). A matrix and a matrix both hold 6 elements, but they are completely different matrices.
Note
In this chapter we write for a matrix of order , and we work only with matrices whose elements are real numbers or functions taking real values.
A point of the plane can be stored as a matrix too — as a column or as a row . So the point becomes or .
Put several points side by side, and a whole figure fits in one matrix. The quadrilateral ABCD below has vertices A(1, 0), B(3, 2), C(1, 3) and D(−1, 2). With one vertex in each column it becomes the matrix
Column 2 of X is the point B(3, 2), and column 4 is D(−1, 2). With one vertex in each row instead, we get a
This is exactly how a computer keeps track of a shape. To rotate, enlarge or reflect it, the computer combines X with one small matrix, and every corner moves at once.
Example 1 : The table shows the number of men and women workers in three factories, I, II and III.
| Factory | Men workers | Women workers |
|---|---|---|
| I | 30 | 25 |
| II | 25 | 31 |
| III | 27 | 26 |
Represent this information as a matrix. What does the element in the third row and second column represent?
Solution : With the factories as rows and men and women as columns,
The element sits in the row of factory III and the column of women workers, so it is the number of
Example 2 : If a matrix has 8 elements, what are the possible orders it can have?
Play first. Choose a number of rows, and the 8 elements try to line up in equal rows. Find every order that works.
Solution : An matrix has elements, so we need every ordered pair of natural numbers whose product is 8: (1, 8), (8, 1), (4, 2) and (2, 4). Hence the possible orders are , , and .
Tip
Listing orders is listing factor pairs. A prime number of elements, such as 13, leaves only
Example 3 : Construct a matrix whose elements are given by .
Solution : A matrix has the elements . Put each and into the rule:
, and
So the required matrix is .
Speed Round: Read the Matrix
Name the order, or read off the element asked for. Rows first, always.
Get five right to move on.