If we have to convert a system into a matrix, check what is placed in rows and columns. The operations we can meaningfully carry out on matrices rely on us understanding the context.
If we need to multiply, set up your matrices such that the dimensions allow multiplication and that the multiplication makes sense in the context of the question.
Always check that the answer seems reasonable.

Matrices allow us to perform multiple individual calculations in one operation and can be used on very large arrays of data. As such, matrix applications are wide reaching, let's look at some further applications.

Examples

Example 1

A second-hand bookstore sells textbooks at a markup of 50\%. The table shows the amounts they paid for old textbooks during the past academic year.

	Semester 1	Semester 2
Business	\$940	\$980
Law	\$1020	\$1170
Mathematics	\$930	\$1160
Science	\$1180	\$1040
Engineering	\$1150	\$970

Organise the purchase costs into a cost matrix, with each row representing a subject and columns representing semesters. C=\begin{bmatrix} ⬚&⬚\\ ⬚&⬚\\ ⬚&⬚\\ ⬚&⬚\\ ⬚&⬚\\ \end{bmatrix}

Worked Solution

Create a strategy

Write the numbers in the matrix in the same position as they are in the table.

Apply the idea

C= \begin{bmatrix} 940&980\\ 1020&1170\\ 930&1160\\ 1180&1040\\ 1150&970\\ \end{bmatrix}

Organise the revenue that will be generated when they manage to sell all the textbooks into a revenue matrix. R=⬚\begin{bmatrix} 940&980\\ 1020&1170\\ 930&1160\\ 1180&1040\\ 1150&970\\ \end{bmatrix}

Worked Solution

Create a strategy

Use the markup percentage as a scalar and perform scalar multiplication.

Apply the idea

A markup of 50\% means that textbooks will be sold at 150\% of the prices the bookstore purchased them at. So we need to multiply each price by 1.5.

\displaystyle R	\displaystyle =	\displaystyle 1.5\begin{bmatrix} 940&980\\ 1020&1170\\ 930&1160\\ 1180&1040\\ 1150&970\\ \end{bmatrix}	Multiply matrix C by 1.5
	\displaystyle =	\displaystyle \begin{bmatrix} 1.5\times 940&1.5\times 980\\ 1.5\times 1020& 1.5 \times 1170\\ 1.5\times 930&1.5\times 1160\\ 1.5\times 1180&1.5\times 1040\\ 1.5\times 1150&1.5\times 970\\ \end{bmatrix}	Multiply each element by 1.5
	\displaystyle =	\displaystyle \begin{bmatrix} 1410&1470\\ 1530&1755\\ 1395&1740\\ 1770&1560\\ 1725&1455\\ \end{bmatrix}	Evaluate

Complete the profit matrix. P= \begin{bmatrix} 1410&1470\\ 1530&1755\\ 1395&1740\\ 1770&1560\\ 1725&1455\\ \end{bmatrix} - \begin{bmatrix} 940&980\\ 1020&1170\\ 930&1160\\ 1180&1040\\ 1150&970\\ \end{bmatrix}

Worked Solution

Create a strategy

Subtract corresponding elements in the two matrices.

Apply the idea

\displaystyle P	\displaystyle =	\displaystyle \begin{bmatrix} 1410-940&1470-980\\ 1530-1020&1755-1170\\ 1395-930&1740-1160\\ 1770-1180&1560-1040\\ 1725-1150&1455-970 \end{bmatrix}	Subtract corresponding elements
	\displaystyle =	\displaystyle \begin{bmatrix} 470&490\\ 510&585\\ 465&580\\ 590&520\\ 575&485 \end{bmatrix}	Evaluate

How much profit would the bookstore have generated from the sale of all these textbooks?

Worked Solution

Create a strategy

Add all the elements in the profit matrix.

Apply the idea

The profit matrix shows the profit for each subject's books per semester. To find the total profit, we add all the elements.

\displaystyle P	\displaystyle =	\displaystyle 470+490+510+585+465+580+590+520+575+485	Add all elements
	\displaystyle =	\displaystyle \$5270	Evaluate

Example 2

The matrix H shows three friends and their scores over four games played against each other. H=\begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} \text{Game } 1 & \text{Game } 2 & \text{Game } 3 & \text{Game } 4 \end{matrix} \\ \begin{matrix} \text{Mandy} \\ \text{Millie} \\ \text{Max} \end{matrix} & \begin{bmatrix} 10\qquad&12\qquad&15\qquad&20 \\ 8\qquad&20\qquad&6\qquad&14\\ 11\qquad&16\qquad&15\qquad&18 \end{bmatrix} \end {matrix}

If A and B are matrices of ones as shown, answer the following questions. A=\begin{bmatrix} 1&1&1 \end{bmatrix}, B=\begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}

Which calculation will give the average number of points that each player scored?

\dfrac{1}{3}AH

\dfrac{1}{4}AH

\dfrac{1}{3}HB

\dfrac{1}{4}HB

Worked Solution

Create a strategy

Find the total number of points each player scored and divide the totals by the number of games.

Apply the idea

To find the total points each player scores we need to find the sum of the elements in each row. This can be done by multiplying each of the 4 elements in each row by 1. So we need to find HB.

To divide by the number of games we need to divide by 4 which is the same as multiplying by \dfrac{1}{4}.

The calculation \dfrac{1}{4}HB will give the average number of points that each player scored. The answer is Option D.

Who had the highest average score overall?

Worked Solution

Create a strategy

Find \dfrac{1}{4}HB to find the average for each player and compare the elements of the result.

Apply the idea

\displaystyle \dfrac{1}{4}HB	\displaystyle =	\displaystyle \dfrac{1}{4} \begin{bmatrix} 10&12&15&20\\ 8&20&6&14\\ 11&16&15&18 \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}	Substitute the matrices
	\displaystyle =	\displaystyle \dfrac{1}{4} \begin{bmatrix} 10+12+15+20\\ 8+20+6+14\\ 11+16+15+18\\ \end{bmatrix}	Multiply and add the elements
	\displaystyle =	\displaystyle \dfrac{1}{4} \begin{bmatrix} 57\\ 48\\ 60 \end{bmatrix}	Evaluate each element
	\displaystyle =	\displaystyle \begin{bmatrix} 14.25\\ 12\\ 15 \end{bmatrix}	Multiply each element by \dfrac{1}{4}

Since the highest average score is 15, Max is the player who had the highest average score overall as he is on the third row of matrix H.

Idea summary

Key things to remember when solving problems using matrices:

If we have to convert a system into a matrix, check what is placed in the rows and columns so we can interpret the results in context.
If we need to multiply, set up your matrices such that the dimensions allow multiplication and that the multiplication makes sense in the context of the question.
Always check that the answer seems reasonable.

Outcomes

U1.AoS3.5

simple communication matrices and their application

U1.AoS3.10

use matrix sums, difference, products and powers and inverses to model and solve practical problems

6.07 Applications of matrices

Ideas

Applications of matrices

Examples

Example 1

Example 2

Outcomes

U1.AoS3.5

U1.AoS3.10

What is Mathspace

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