Topics
14. Matrices and Networks
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Middle Years

14.03 Matrix multiplication

Lesson
Worksheet
Practice
Worksheet
Multiplying matrices
1

For each pair of matrices below:

i
State whether the product AB is defined. If so, answer parts (ii) and (iii).
ii
State the dimensions of AB.
iii
Find the matrix AB.
a
A = \begin{bmatrix} 8 \end{bmatrix} \text{ and } B = \begin{bmatrix} 4 \end{bmatrix}
b
A = \begin{bmatrix} 6 & 5 \\ -3 & -7 \\ 9 & 2 \end{bmatrix} \text{ and } B = \begin{bmatrix} -8 \\ 3 \end{bmatrix}
c
A = \begin{bmatrix} 5 & -9 & 4 & -2 \end{bmatrix} \text{ and } B = \begin{bmatrix} 9 \\ -7 \\1 \\ -5 \end{bmatrix}
d
A = \begin{bmatrix} -7 \\ 7 \end{bmatrix} \text{ and }B = \begin{bmatrix} 3 \\ -6 \end{bmatrix}
e
A = \begin{bmatrix} 3 & 2 \\ 9 & 6 \end{bmatrix} \text{ and } B = \begin{bmatrix} -7 & -4 \\ -5 & -6 \end{bmatrix}
f
A = \begin{bmatrix} 7 \\ 1 \end{bmatrix} \text{ and } B = \begin{bmatrix} 4 & 9 -5 \end{bmatrix}
g
A = \begin{bmatrix} 9 & -5 & -6 \\ 6 & -8 & 5 \end{bmatrix} \text{ and } B = \begin{bmatrix} -4 & -1 & 8 \\ -3 & -9 & 4 \end{bmatrix}
h
A = \begin{bmatrix} 8 & -6 & 5 \\ 7 & -8 & -9 \\ -5 & -7 & 1 \end{bmatrix} \text{ and } B = \begin{bmatrix} 2 \\ 9\\ -3 \end{bmatrix}
i
A = \begin{bmatrix} 0 & 0 & 5 & 5 \\ 2 & -5 & 2 & -4 \\ 1 & 8 & 3 & 6 \end{bmatrix} \text{ and } B = \begin{bmatrix} 9 & 1 \\ 8 & -2 \\ 7 & 4 \\ 7 & -1 \end{bmatrix}
j
A = \begin{bmatrix} -2 & -3 \\ 7 & 6 \end{bmatrix} \text{ and } B = \begin{bmatrix} 4 & 5 & -1 \\ -7 & 1 & 9 \end{bmatrix}
k
A = \begin{bmatrix} 7 & 9 & -3 \\ -8 & -7 & -6 \end{bmatrix} \text{ and } B = \begin{bmatrix} -9 & 1 \\ 6 & -2 \\ -5 & 4 \end{bmatrix}
l
A = \begin{bmatrix} -5 & 5 \\ 3 & -2 \\ -3 & 8 \end{bmatrix} \text{ and } B = \begin{bmatrix} 2 & 6 & 4 \\ -6 & 1 & -1 \end{bmatrix}
m
A = \begin{bmatrix} 3 & -2 \\ 8 & -3 \\ 4 & -4 \\ -1 & 5 \end{bmatrix} \text{ and } B = \begin{bmatrix} -6 & 6 & 1 \\ 2 & 7 & -5 \end{bmatrix}
n
A = \begin{bmatrix} -6 & 2 & 4 \end{bmatrix} \text{ and } B = \begin{bmatrix} 1 \\ 7 \\ -3 \end{bmatrix}
2

Consider A = \begin{bmatrix} 7 & 9 & -2 \\ 4 & 1 & -3 \end{bmatrix}. If A is to be multiplied by B, a column matrix, what are the dimensions of matrix B?

3

If A is a 4 \times 2 matrix and the product AB is a 4 \times 5 matrix, what are the dimensions of B?

4

A matrix calculation of A = BC resulted in the matrix A = \begin{bmatrix} 6 & -2 & 3 \\ 2 & 1 & -3 \\ -5 & 8 & -1 \end{bmatrix}. If B is a 3 \times 2 matrix, what are the dimensions of C?

5

The matrix A = \begin{bmatrix} 6 & -2 & 3 \\ 2 & 1 & -3 \\ -5 & 8 & -1 \end{bmatrix} is used in the matrix multiplication BAC where BAC is a 3 \times 3 matrix.

a
What are the dimensions of B?
b
What are the dimensions of C?
6

Consider the matrices A =\begin{bmatrix} 2 & 3 & 7 \\ -6 & -4 & 9 \end{bmatrix} and B = \begin{bmatrix} 0 & -3 \\ -1 & 8 \\ 6 & -2 \end{bmatrix}.

a
What are the dimensions of AB?
b
Find the matrix AB.
c
What are the dimensions of BA?
d
Find the matrix BA.
e
Is AB equal to BA?
7

Solve the following equations for n:

a
\begin{bmatrix} -3 & n \end{bmatrix} \begin{bmatrix} 7 \\ -9 \end{bmatrix} = \begin{bmatrix} -57 \end{bmatrix}
b
\begin{bmatrix} 8 & -5 & -6 \end{bmatrix} \begin{bmatrix} 7 \\ n \\ 4 \end{bmatrix} = \begin{bmatrix} 7 \end{bmatrix}
c
\begin{bmatrix} 2 & 9 & 3 \\ 4 & -2 & n \end{bmatrix} \begin{bmatrix} 8 \\ 6 \\ -4 \end{bmatrix} = \begin{bmatrix} 58 \\ 32 \end{bmatrix}
Applications
8

Tobias owns two pizza stores, Paris Pizza and Penrith Pizza, at which he sells small pizzas for \$5, and medium-sized pizzas for \$11 and large pizzas for \$18. The table shows the number of pizzas sold at each store on a particular day:

a
Organise the prices into a 3 \times 1 column matrix, A, in increasing size order.
b
Organise the number of pizzas sold into a 2 \times 3 matrix, B, as given in the table.
c
Calculate Tobias's total revenue for each store by finding BA.
SmallMediumLarge
Paris Pizza261128
Penrith Pizza192435
9

A local bakery was selling three different products yesterday. The table shows the price that each product was sold for and the amount sold:

a
Organise the prices of each product into a 1 \times 3 row matrix, A, in the order given in the table.
b
Organise the quantity sold of each product into a 3 \times 1 column matrix, B, in the order given in the table.
c
Calculate the bakery's total revenue for the day by finding AB.
PriceQuantity Sold
Sandwich\$622
Pork roll\$317
Bread roll\$721
10

In the last cricket season, the Sydney Sickies had 13 wins, 2 ties, 5 draws and 8 losses. The table shows the points system used in the competition:

a
Construct the team's results in each cricket match into a 1 \times 4 row matrix A. Order the elements from left to right as the number of wins, ties, draws and losses.
b
Construct the points system into a 4 \times 1 column matrix B. Order the elements from top to bottom as the points for each win, tie, draw and loss.
c
Calculate the team's total points for the season by finding AB.
Points
Win6
Tie3
Draw2
Loss0
11

Fred's Fast Food sells three different meals. The table shows the number of items they bundle together into each type of meal.

HamburgersOnion RingsChicken NuggetsPizzasSoft Drinks
Mega Meal591204
Hungry Meal38513
Crazy Meal25941

On Saturday, they sold 25 Mega Meals (M), 22 Hungry Meals (H) and 24 Crazy Meals (C). On Sunday, they sold 16 Mega Meals, 19 Hungry Meals and 21 Crazy Meals.

a
Organise the number of meals sold each day into a 2 \times 3 matrix A. Order the columns by Saturday then Sunday, and rows by M, H \text{ then } C.
b
Organise the number of items in each meal into a 3 \times 5 matrix B as given in the table.
c
Find AB.
d
How many hamburgers did Fred's Fast Food sell on Sunday?
e
How many pizzas did Fred's Fast Food sell on the weekend?
12

A movie theatre has special charity screenings of a new movie over two nights. On Friday 80 adults and 25 children attended, and on Saturday 195 adults and 51 children attended. Adult tickets are priced at \$20 and child tickets are priced at \$15.

a
Organise the number of adults and children that attended the movie on Friday and Saturday into a 2 \times 2 matrix, where the rows are Friday then Saturday and the columns are adults then children.
b
Organise the prices in a 2 \times 1 matrix, in order of adults then children.
c
Multiply your matrices to find the total amount spent at the movie theatre on each night.
d
If 15\% of ticket sales for the event are donated to charity, determine the total that will be given to charity over the two nights.
13

The quarterly mobile phone bills for three friends in 2019 are represented in the table below:

Quarter 1Quarter 2Quarter 3Quarter 4
Sharon80879082
Ray76757374
Neil45505246
a
Organise the data from the table into a 3 \times 4 matrix, P. Use the same order as in the table.
b
Let A = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, and calculate PA.
c
Let B = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}, and calculate BP.
d
Which matrix product represents the total yearly bill for each person: PA \text{ or } BP?
e
State what the other matrix product represents.
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