Applications of Matrices Revision Notes for VCE SSCE General Mathematics

Applications of Matrices

Introduction

Matrices provide a powerful way to organize and process data. When information is stored in matrix form, we can use matrix multiplication to extract new, useful results from the original data. This technique is particularly valuable in business contexts, data analysis, and situations where we need to combine different types of information efficiently.

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Matrix multiplication isn't just a mathematical operation—it's a practical tool that allows us to combine different datasets and calculate meaningful totals in a single, efficient operation. This makes matrices essential in business analytics, scientific research, and data processing.

Business applications of matrices

Calculating sales and promotions

Matrices are commonly used in business to combine different types of data and calculate totals. When we organize sales quantities and price information into separate matrices, multiplying them together automatically calculates total revenue and other business metrics.

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Worked Example: Store Sales with Promotions

Scenario: A store runs a promotion where customers receive free cinema tickets with their purchases. We need to calculate total sales revenue and the number of tickets distributed.

Setting up the problem:

We use two matrices to organize our information:

Matrix $S$ represents sales data (who sold what):

Rows represent different salespeople
Columns represent different products

Matrix $P$ represents product information:

Rows represent different products
Columns represent price and promotional items

The specific case:

Freddy and George work at a store selling DVDs and games. The promotion gives:

One free cinema ticket with each DVD purchased
Two free cinema tickets with each game purchased

The sales data is organized in matrix $S$ :

$S = \begin{bmatrix} 7 & 4 \\ 5 & 6 \end{bmatrix}$

where rows represent Freddy and George, columns represent DVDs and Games.

The price and ticket data is organized in matrix $P$ :

$P = \begin{bmatrix} 20 & 1 \\ 30 & 2 \end{bmatrix}$

where rows represent DVDs and Games, columns represent price ($) and tickets.

Calculating the results:

To find total sales and tickets distributed, we multiply $S \times P$ :

$S \times P = \begin{bmatrix} 7 & 4 \\ 5 & 6 \end{bmatrix} \times \begin{bmatrix} 20 & 1 \\ 30 & 2 \end{bmatrix}$

Step-by-step multiplication:

First row (Freddy's results):

Sales: $7 \times 20 + 4 \times 30 = 140 + 120 = 260$
Tickets: $7 \times 1 + 4 \times 2 = 7 + 8 = 15$

Second row (George's results):

Sales: $5 \times 20 + 6 \times 30 = 100 + 180 = 280$
Tickets: $5 \times 1 + 6 \times 2 = 5 + 12 = 17$

$S \times P = \begin{bmatrix} 260 & 15 \\ 280 & 17 \end{bmatrix}$

Interpreting the results:

The resulting matrix tells us:

Freddy had total sales of $260 and distributed 15 cinema tickets
George had total sales of $280 and distributed 17 cinema tickets

This demonstrates how matrix multiplication combines sales quantities with prices and promotion details to calculate meaningful business totals in a single operation.

Practice opportunity: Try calculating similar results for Jacky and Peter's store, which sells drinks ($4 each, 1 ticket) and hamburgers ($12 each, 2 tickets), where Jacky sold 3 drinks and 10 hamburgers, while Peter sold 6 drinks and 8 hamburgers.

Extracting information using row and column matrices

Understanding row and column matrices

Row and column matrices are special matrices that contain just one row or one column. These simple matrices have a powerful property: they can efficiently extract summary information from larger data matrices.

Key definitions:

A row matrix has dimensions $1 \times n$ (one row, multiple columns)
A column matrix has dimensions $m \times 1$ (multiple rows, one column)

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The beauty of row and column matrices is their simplicity. By filling them with 1s, we create powerful tools for calculating totals. A column matrix of 1s acts like a "sum each row" command, while a row matrix of 1s acts like a "sum each column" command.

Using column matrices to calculate row sums

When we multiply a data matrix by a column matrix containing all 1s (on the right side), the result gives us the sum of each row in the original matrix. This operation efficiently calculates row totals without manual addition.

Using row matrices to calculate column sums

When we multiply a row matrix containing all 1s by a data matrix (on the left side), the result gives us the sum of each column in the original matrix. This provides quick access to column totals across the entire dataset.

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Worked Example: Wildlife Survey Data

Scenario: Three park rangers survey feral animal sightings monthly and record their observations.

The sightings data is stored in matrix $S$ :

$S = \begin{bmatrix} 27 & 9 & 34 & 59 \\ 18 & 15 & 10 & 89 \\ 35 & 6 & 46 & 29 \end{bmatrix}$

where rows represent Aaron, Barry, and Chloe, and columns represent cats, dogs, foxes, and rabbits.

We'll use two special matrices:

$A = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}$ (row matrix)

$B = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}$ (column matrix)

Part 1: Finding row sums with a column matrix

Calculate $S \times B$ :

$S \times B = \begin{bmatrix} 27 & 9 & 34 & 59 \\ 18 & 15 & 10 & 89 \\ 35 & 6 & 46 & 29 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}$

Working through each row:

$= \begin{bmatrix} 27 + 9 + 34 + 59 \\ 18 + 15 + 10 + 89 \\ 35 + 6 + 46 + 29 \end{bmatrix}$

$= \begin{bmatrix} 129 \\ 132 \\ 116 \end{bmatrix}$

Interpretation: Each element in $S \times B$ gives the sum of the corresponding row in $S$ . This tells us the total number of animal sightings made by each ranger:

Aaron: 129 total sightings
Barry: 132 total sightings
Chloe: 116 total sightings

Part 2: Finding column sums with a row matrix

Calculate $A \times S$ :

$A \times S = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \times \begin{bmatrix} 27 & 9 & 34 & 59 \\ 18 & 15 & 10 & 89 \\ 35 & 6 & 46 & 29 \end{bmatrix}$

Working through each column:

$= \begin{bmatrix} 27+18+35 & 9+15+6 & 34+10+46 & 59+89+29 \end{bmatrix}$

$= \begin{bmatrix} 80 & 30 & 90 & 177 \end{bmatrix}$

Interpretation: Each element in $A \times S$ gives the sum of the corresponding column in $S$ . This tells us the total number of sightings for each type of animal across all rangers:

Cats: 80 total sightings
Dogs: 30 total sightings
Foxes: 90 total sightings
Rabbits: 177 total sightings

Why this works:

When we multiply by a column matrix of 1s on the right, each element in the result equals one row from the original matrix multiplied by all 1s, which is simply the sum of that row.

When we multiply by a row matrix of 1s on the left, each element in the result equals the row of 1s multiplied by one column from the original matrix, which is simply the sum of that column.

This provides an efficient way to calculate totals without manually adding up rows or columns.

chatImportant

Remember the pattern: Column matrix on the RIGHT gives ROW sums ( $S \times B$ ), and row matrix on the LEFT gives COLUMN sums ( $A \times S$ ).

A common mistake is to confuse which position gives which type of sum. Think of it this way: the column matrix is tall (like columns), but it sums the rows when placed on the right!

Practice opportunity: Try applying this technique to calculate total TV watching time for four students (Beth, Zara, Aria, Wanda) across five weekdays (Monday through Friday).

Summary

Matrices provide efficient methods for organizing, combining, and analyzing data. Matrix multiplication allows us to:

Combine different types of information (such as sales quantities and prices) to calculate totals
Use row and column matrices to quickly extract row sums and column sums from large data sets
Perform multiple calculations simultaneously in a single matrix operation

These techniques are particularly valuable in business applications, data analysis, and any situation where structured data needs to be processed efficiently.

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Key Points to Remember:

When organizing data in matrices, matrix multiplication can combine information and produce useful results
In business applications, multiplying sales data by price data calculates total revenue
A column matrix of 1s multiplied on the right ( $S \times B$ ) gives the sum of each row
A row matrix of 1s multiplied on the left ( $A \times S$ ) gives the sum of each column
Always interpret your results in context—what do the numbers in the resulting matrix actually represent?

Applications of Matrices (VCE SSCE General Mathematics): Revision Notes