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

State the dimensions of matrix A.

State the dimensions of matrix B.

Is A + B possible?

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

State the dimensions of matrix A.

State the dimensions of matrix B.

State the dimensions of matrix C.

Is A + B possible?

Is B - C possible?

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}

State the dimensions of matrix A.

State the dimensions of matrix B.

State the dimensions of matrix C.

Is A + B possible?

Is A + B - C possible?

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

State the dimensions of matrix A.

State the dimensions of matrix B.

State the dimensions of matrix C.

Is C - B possible?

Is C + A - B possible?

Find A + B, if:

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

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

A = \begin{bmatrix} 1 & \frac{5}{2} & 8 \\ 5 & -7 & 10 \end{bmatrix} and B = \begin{bmatrix} 2 & \frac{3}{2} & 9 \\ -3 & 0 & 7 \end{bmatrix}

A = \begin{bmatrix} 4 & 8 \\ 5 & 10 \\ -2 & 6 \end{bmatrix} and B = \begin{bmatrix} 1 & 9 \\ -4 & 2 \\ 3 & -1 \end{bmatrix}

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}

Find A - B, if:

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

A = \begin{bmatrix} -2 \\ 4 \\ 3 \end{bmatrix} and B = \begin{bmatrix} 8 \\ 0 \\ -3 \end{bmatrix}

A = \begin{bmatrix} -\frac{1}{3} & 7 & 12 \\ 11 & -3 & 0 \end{bmatrix} and B = \begin{bmatrix} \frac{2}{3} & 1 & 0 \\ 4 & 8 & 7 \end{bmatrix}

A = \begin{bmatrix} 6 & 7 \\ 9 & 10 \\ 4 & -6 \end{bmatrix} and B = \begin{bmatrix} 2 & 5 \\ -1 & 3 \\ 6 & 1 \end{bmatrix}

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}

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}

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

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

-4A

\dfrac{1}{2}A

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

-A

10A

\dfrac{1}{2}A

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

4A + B

2A - B

5A + 3B

4A - 2B

\dfrac{1}{2}A + \dfrac{1}{4}B

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

4A + 2B

2A - 2B

3A + 5B

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

Matrix equations

Solve the following matrix equations for x:

\begin{bmatrix} 5 & x \\ -2 & 9 \end{bmatrix} + \begin{bmatrix} 7 & 3 \\ 8 & 2 \end{bmatrix} = \begin{bmatrix} 12 & 8 \\ 6 & 11 \end{bmatrix}

\begin{bmatrix} 10 & 8 \\ -4 & 10 \end{bmatrix} - \begin{bmatrix} 8 & 3 \\ 2x & 7 \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ -14 & 3 \end{bmatrix}

\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}

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:

Solve for n:

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}

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}

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}

Solve the following matrix equations for matrix A:

2 \begin{bmatrix} 8 & 4 \\ -2 & 5 \end{bmatrix} + A = \begin{bmatrix} 22 & 18 \\ 10 & -8 \end{bmatrix}

4 \begin{bmatrix} 3 & -1 \\ 6 & 0 \end{bmatrix} - 2A = \begin{bmatrix} 14 & 8 \\ 12 & -24 \end{bmatrix}

Applications

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

	Australia	New Zealand	Thailand	China
January	37	25	29	41
February	26	35	19	51
March	32	22	18	27
April	30	28	24	37
May	31	20	24	28

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.

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.

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

Saturday:

	Fruit	Vegetables
Shop 1	58	29
Shop 2	48	71
Shop 3	54	38

Sunday:

	Fruit	Vegetables
Shop 1	45	32
Shop 2	40	62
Shop 3	38	46

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

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

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.

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 sales price of the four products and write them in a 1 \times 4 matrix.

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

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.

Organise the percentages into a 1 \times 4 row matrix in the same order as stated above.

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

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

Using scalar product, find the marked up prices. Express your answer as a 3 \times 2 matrix.

	Pizza	Drinks
Small	\$6	\$3
Medium	\$8	\$4.50
Large	\$12	\$6

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

Using scalar product, find the discounted prices. Express your answer as a 3 \times 2 matrix.

	Small	Large
Sandwiches	\$4.50	\$7
Pies	\$5	\$8.50
Cakes	\$6	\$9.20

14.02 Matrix addition, subtraction and scalar multiplication

What is Mathspace

About Mathspace