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Mathematical Operations on Matrices 1

What you need to know

- General algebraic methods
- Use of functions
- Some knowledge of vectors and how to manipulate

them - The content of Introduction to Matrices.
- IMPORTANT NOTE MAKE SURE YOU ARE COMFOTABLE WITH

THE ABOVE BEFORE PROCEEDING WITH THIS SECTION.

MATHS AND MATRICES

- This section will cover some simple mathematical

operations using matrices - Addition (covered in part 1)
- Subtraction (covered in part 1)
- Multiplication by a scalar (covered in part 2)
- Multiplication by another matrix (covered in part

2) - The Inverse Matrix (covered in part 3)
- It will also give practical examples and

exercises in representing and manipulating data

in a matrix. - As you may have noticed, this topic has been

split into three separate presentations, entitled

Mathematical Methods and numbered 1 to 3.

These should be worked through in order.

Storing Information

- As you will remember, a matrix is a rectangular

array of numbers, and can be a useful way of

storing information. - Hopefully you remember matrix A
- 1 3 5 7 A 2 4 6

8 5 9 0 2 - In this form it could be storing any information.

In this respect the matrix could be interpreted

differently depending on the situation. The

structure does not decide the content, and this

should be borne in mind as we use this matrix.

Storing Information 2

- Previously, we considered that the matrix was an

example of sales data from a hardware store. We

decided that it showed the sales of three

products over the course of four days. A was

considered as a form of table, in which we have

stored the following information Mon

Tues Wed Thurs PANS 1 3 5

7 NAILS 2 4 6 8 SAWS 5

9 0 2

Addition using Matrices

- Suppose that we had a companion store in another

town, such that one is in town 1 and the other in

town 2, and we would like to be able to calculate

the total sales of the items in both outlets. - We create a matrix of the data from the second

shop in a matrix of the same kind as the first.

This gives us matrix B. - 1 3 5 7 2 4 4 9 A 2 4

6 8 B 3 3 7 7 5 9 0 2 8 6

4 1 - To add two matrices of the same kind, we can

simply add the corresponding elements, as we

shall see.

Addition with Matrices 2

- 1 3 5 7 2 4 4 9 A B 2

4 6 8 3 3 7 7 5 9 0 2

8 6 4 1 - 12 34 54 79 A B 23

43 67 87 58 96 04 21 - 3 7 9 16 A B 5 7 13

15 C 13 15 4 3 - Can you understand how this works?

Summary of Addition with Matrices

- If two matrices are of the same kind, they can be

added together to produce a new matrix of the

same kind. - This means that if Aai j and Bbi j are both

of the same kind the following can be presumed - C A B ai j bi j
- It should also be noted that matrices have

commutative properties. This means that in the

case of matrix addition A B B A. In

simple terms, this means that for addition, the

order of manipulation, or the order of the

matrices, does not affect the result (as long as

they are added correctly!).

Subtraction of Matrices

- Subtraction of matrices works much the same as

addition. Suppose we want to find out whether

the first shop, represented by A, sold more of

the items than shop B. - 1 3 5 7 2 4 4 9 A 2 4

6 8 B 3 3 7 7 5 9 0 2 8 6

4 1 - How would you do this? Try to work it out and

then go to the next page for a worked example.

Subtraction of Matrices 2

- 1 3 5 7 2 4 4 9 A- B 2

4 6 8 - 3 3 7 7 5 9 0 2

8 6 4 1 - 1-2 3-4 5-4 7-9 A -B 2-3

4-3 6-7 8-7 5-8 9-6 0-4 2-1 - -1 -1 1 -2 A -B -1 1

-1 1 D -3 3 -4 1 - Can you see how this works? In this case A does

not seem to have sold much more than B, and may

have possibly sold less on average.

Subtraction of Matrices 3

- The best way to find the differences between two

matrices of the same kind is actually using

addition, by using the Opposite Matrix of the

matrix we want to subtract with A B

A -B -B A - We calculate the opposite matrix, by changing the

signs, and then add that to the matrix we want to

subtract from. - NB. It may seem unnecessarily long-winded, but it

is less prone to mistakes, and easier to check

for errors.

Exercise 2 See Answers.ppt

- Add the following matrices, where possible
- 5 7 6 8
- 10 12 8 12 15 17

9 42 7 15 4

13 - 9 8 7 6 5 4 4 5 6

7 8 9 1 3 2

Exercise 2 Continued

- Subtract the following matrices, where possible
- 5 - 7 6 8
- 10 12 - 8 12 15 17

9 42 7 15 4 13 - 9 8 7 - 6 5 4 4 5 6

7 8 9 1 3 2 - -9 8 7 - -6 5 4 4 -5 6

7 -8 -9 1 -3 2

Summary

- In this section, you have learnt how to

manipulate matrices to perform the simple

mathematical operations of addition and

subtraction. - Please make sure that you are clear on the topics

that have been covered in this part before

proceeding to the next part of this course

Mathematical Operations on Matrices 2, where

you will be shown how to multiply a matrix by a

scalar, and how to multiply two matrices

together. - Remember that the answers to the set questions

can be found in Answers.ppt.

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