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4. Algebra and Matrices
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Year 11

4.05 Matrix addition, subtraction and scalar multiplication

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Worksheet
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Worksheet
Matrix addition and subtraction
1

Consider each pair of matrices:

i

State the dimensions of matrix A.

ii

State the dimensions of matrix B.

iii

Is A + B possible?

a

A = \begin{bmatrix} 1 & 5 \\ 3 & 2 \end{bmatrix} and B = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}

b

A = \begin{bmatrix} 1 & 6 \\ -2 & 2 \\ 8 & 0 \end{bmatrix} and B = \begin{bmatrix} 3 & -1 & 5 \\ 2 & 4 & 7 \end{bmatrix}

2

Consider the matrices A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, B = \begin{bmatrix} a \\ b \end{bmatrix} and C = \begin{bmatrix} a & b \end{bmatrix}.

a

State the dimensions of matrix A.

b

State the dimensions of matrix B.

c

State the dimensions of matrix C.

d

Is A + B possible?

e

Is B - C possible?

3

Consider the matrices:

A = \begin{bmatrix} 1 & 5 & 3 \\ 2 & -1 & 6 \\ 10 & 7 & 8 \end{bmatrix} \text {, } B = \begin{bmatrix} 2 & -4 & 1 \\ 8 & 3 & -9 \\ 11 & 12 & 0 \end{bmatrix} \text{and } C = \begin{bmatrix} 1 & -6 \\ 5 & 9 \end{bmatrix}
a

State the dimensions of matrix A.

b

State the dimensions of matrix B.

c

State the dimensions of matrix C.

d

Is A + B possible?

e

Is A + B - C possible?

4

Consider the matrices A = B = C = \begin{bmatrix} a & b \\ c & d \\ e & f \\ g & h \end{bmatrix}.

a

State the dimensions of matrix A.

b

State the dimensions of matrix B.

c

State the dimensions of matrix C.

d

Is C - B possible?

e

Is C + A - B possible?

5

Find A + B, if:

a
A = \begin{bmatrix} 2 & 5 \\ 4 & 3 \end{bmatrix} and B = \begin{bmatrix} 5 & 4 \\ -1 & 6 \end{bmatrix}
b
A = \begin{bmatrix} -1 \\ 5 \\ 7 \end{bmatrix} and B = \begin{bmatrix} 2 \\ 4 \\ -3 \end{bmatrix}
c
A = \begin{bmatrix} 1 & 5 & 8 \\ 5 & -7 & 10 \end{bmatrix} and B = \begin{bmatrix} 2 & 3 & 9 \\ -3 & 0 & 7 \end{bmatrix}
d
A = \begin{bmatrix} 4 & 8 \\ 5 & 10 \\ -2 & 6 \end{bmatrix} and B = \begin{bmatrix} 1 & 9 \\ -4 & 2 \\ 3 & -1 \end{bmatrix}
e
A = \begin{bmatrix} 5 & 2 & 8 \\ 6 & 10 & -3 \\ 1 & -2 & 6 \end{bmatrix} and B = \begin{bmatrix} 1 & 5 & 9 \\ 4 & 6 & -2 \\ 1 & 3 & -1 \end{bmatrix}
6

Find A - B, if:

a
A = \begin{bmatrix} 1 & 4 \\ 5 & 2 \end{bmatrix} and B = \begin{bmatrix} 8 & 0 \\ -1 & 7 \end{bmatrix}
b
A = \begin{bmatrix} -2 \\ 4 \\ 3 \end{bmatrix} and B = \begin{bmatrix} 8 \\ 0 \\ -3 \end{bmatrix}
c
A = \begin{bmatrix} -1 & 7 & 12 \\ 11 & -3 & 0 \end{bmatrix} and B = \begin{bmatrix} 6 & 1 & 0 \\ 4 & 8 & 7 \end{bmatrix}
d
A = \begin{bmatrix} 6 & 7 \\ 9 & 10 \\ 4 & -6 \end{bmatrix} and B = \begin{bmatrix} 2 & 5 \\ -1 & 3 \\ 6 & 1 \end{bmatrix}
e
A = \begin{bmatrix} 7 & 4 & 2 \\ 11 & 10 & 8 \\ 9 & 3 & 7 \end{bmatrix} and B = \begin{bmatrix} 5 & -3 & 1 \\ 5 & 3 & 2 \\ 0 & 4 & -1 \end{bmatrix}
f
A = \begin{bmatrix} 3.6 & 7.4 \\ 8.9 & 10.5 \\ 4.7 & -6.3 \end{bmatrix} and B = \begin{bmatrix} 1.8 & 5.1 \\ 0.9 & 5.2 \\ -6.1 & 1.5 \end{bmatrix}
g
A = \begin{bmatrix} 75 & 42 & 21 & 54 \\ 18 & 27 & 81 & 56 \\ 93 & 36 & 72 & 49 \end{bmatrix} and B = \begin{bmatrix} 51 & -32 & 18 & 21 \\ 56 & 39 & 24 & 57\\ 18 & 27 & -19 & 32 \end{bmatrix}
Scalar multiplication
7

If A = \begin{bmatrix} 8 & 4 \\ 12 & -10 \end{bmatrix}, find:

a
2A
b
-4A
c
\dfrac{1}{2}A
8

If A = \begin{bmatrix} -7 & 8 & 14 \\ 0 & 15 & 20 \end{bmatrix}, find:

a
3A
b
-A
c
10A
d
\dfrac{1}{2}A
9

If A = \begin{bmatrix} 8 & -3 \\ 2 & 10 \end{bmatrix} and B = \begin{bmatrix} 6 & 10 \\ -1 & 5 \end{bmatrix}, find:

a
4A + B
b
2A - B
c
5A + 3B
d
4A - 2B
e
\dfrac{1}{2}A + \dfrac{1}{4}B
10

If A = \begin{bmatrix} 2.25 & -1.5 \\ 4.5 & 8.25 \end{bmatrix} and B = \begin{bmatrix} 5.5 & 1.25 \\ -1 & 6.5 \end{bmatrix}, find:

a
4A + 2B
b
2A - 2B
c
3A + 5B
11

Let A = \begin{bmatrix} 5 & -4 \\ 3 & 0 \end{bmatrix} \text{, } B = \begin{bmatrix} -10 & 6 \\ 2 & -1 \end{bmatrix} and C = \begin{bmatrix} 8 & 5 \\ -1 & 4 \end{bmatrix}. Find 4A - B + 2C.

Matrix equations
12

Solve the following matrix equations for x:

a
\begin{bmatrix} 5 & x \\ -2 & 9 \end{bmatrix} + \begin{bmatrix} 7 & 3 \\ 8 & 2 \end{bmatrix} = \begin{bmatrix} 12 & 8 \\ 6 & 11 \end{bmatrix}
b
\begin{bmatrix} 10 & 8 \\ -4 & 10 \end{bmatrix} - \begin{bmatrix} 8 & 3 \\ 2x & 7 \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ -14 & 3 \end{bmatrix}
c
\begin{bmatrix} 11 & 7 \\ -x & 10 \\ 4 & 9 \end{bmatrix} - \begin{bmatrix} 6 & 2 \\ -8 & -1 \\ 0 & -3 \end{bmatrix} = \begin{bmatrix} 5 & 5 \\ 9 & 11 \\ 4 & 12 \end{bmatrix}
13

Consider the following matrix equation:

\begin{bmatrix} u - 3 & 2v & 9 \\ 3x & 7 & 10 \end{bmatrix} + \begin{bmatrix} 4u & v & -3w \\ 10 & 5 & 27 \end{bmatrix} = \begin{bmatrix} 47 & 12 & 18 \\ 31 & 6y & 37 \end{bmatrix}

Find the value of:

a
u
b
v
c
w
d
x
e
y
14

Solve the following matrix equations for n:

a
3 \begin{bmatrix} 5 & 2 \\ -1 & 7 \end{bmatrix} + 2 \begin{bmatrix} 4 & -1 \\ n & -6 \end{bmatrix} = \begin{bmatrix} 23 & 4 \\ 17 & 9 \end{bmatrix}
b
4 \begin{bmatrix} 3 & 1 \\ -2 & n \end{bmatrix} - 3 \begin{bmatrix} 5 & 2 \\ -3 & 8 \end{bmatrix} = \begin{bmatrix} -3 & -2 \\ 1 & 40 \end{bmatrix}
c
2 \begin{bmatrix} 8 & 2 \\ -1 & 5 \\ 4 & -3 \end{bmatrix} - \begin{bmatrix} 10 & 1 \\ 6 & -2 \\ 12 & 9 \end{bmatrix} = \begin{bmatrix} 3n & 3 \\ -8 & 12 \\ -4 & -15 \end{bmatrix}
15

Solve the following matrix equations for matrix A:

a
2 \begin{bmatrix} 8 & 4 \\ -2 & 5 \end{bmatrix} + A = \begin{bmatrix} 22 & 18 \\ 10 & -8 \end{bmatrix}
b
4 \begin{bmatrix} 3 & -1 \\ 6 & 0 \end{bmatrix} - 2A = \begin{bmatrix} 14 & 8 \\ 12 & -24 \end{bmatrix}
Applications
16

The following table shows the number of visitors to a website by country:

AustraliaNew ZealandThailandChina
January37252941
February26351951
March32221827
April30282437
May31202428
a

Find the total number of visitors to the site during each month. Express your answer as a column matrix, where the rows describe the months in order of the given table.

b

Find how many more visitors were from China than from Thailand each month by subtracting an appropriate pair of column matrices. Express your answer as a column matrix, where the rows describe the months in order of the given table.

17

The tables below show the number of fruit and vegetables sold at Mohamad's three corner shops over a particular weekend:

Saturday:

FruitVegetables
Shop 15829
Shop 24871
Shop 35438

Sunday:

FruitVegetables
Shop 14532
Shop 24062
Shop 33846
a

Write the sales of fruit and vegetables for Saturday as a 3 \times 2 matrix.

b

Write the sales of fruit and vegetables for Sunday as a 3 \times 2 matrix.

c

Add your matrices to find the total number of sales of fruit and vegetables for each shop over the entire weekend. Express your answers as a 3 \times 2 matrix.

18

The cost matrix for four products at a health store is given by the matrix:

C = \begin{bmatrix} 9.50 & 10.20 & 8.90 & 12.50 \end{bmatrix}

The store adds 120 \% to the cost price to generate the sales price.

Find the profits from each of the four products and write them in a 1 \times 4 matrix.

19

In a particular town, 30\% of households own no pets, 50\% of households own one pet, 15\% of households own two pets and 5\% of households own more than two pets.

If there are 300 households in the town, find the number of households in each category. Express your answer as a 1 \times 4 matrix.

20

A pizzeria is about to mark up prices on their items by 140\%. The table shows their current prices.

Find the marked up prices. Express your answer as a 3 \times 2 matrix.

PizzaDrinks
Small\$6\$3
Medium\$8\$4.50
Large\$12\$6
21

Glorious Jeans will be offering a 25\% discount on all food items for their Boxing Day sales. The table shows their regular prices.

Find the discounted prices. Express your answer as a 3 \times 2 matrix.

SmallLarge
Sandwiches\$4.50\$7
Pies\$5\$8.50
Cakes\$6\$9.20
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2.2.2.3

perform matrix addition, subtraction, and multiplication by a scalar

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