1. Matrices
Tuesday, 23 April 2019

2. Matrices A matrix is a rectangular array of numbers Eg
The size of a matrix is determined by the number of rows and the number of columns it contains
Representation of a matrix is as with vectors, in textbooks Matrix A could by in bold type A and written has matrix A as A
Contains 2 rows and 3 columns and is described as a 2 by 3 (or 2  3) matrix.

3. A matrix with one column is known as a column vectorEg
A matrix with an equal number of rows and columns is known as a square matrix
A Matrix is a mathematical tool which is used in a variety of ways.
Introduced by the Chinese is was first used to solve simultaneous equations.
During the war it was used when building aircraft to deal investigating the aerodynamic s.
In more recent times Matrices are used in designing computer gaming graphics, the latest 3D computer animation
Matrices are used to calculate gross domestic product in economics
Or more importantly used with encryption: The internet function could not function without encryption, and neither could banks since they now use these same means to transmit private and sensitive data

4. Addition and Subtraction of matrices
If two matrices are the same size then they can be added together by summing corresponding entries (or members of the matrices)
Example NB
If the matrices were not the same size then the addition of the two would have no meaning.
If A + B exists the B + A also exists. In fact A + B = B + A

5. Example
Find where possible a) b)
Impossible

6. Multiplying a matrix by a scalar quantity
Example
If find
Here we simply double each entry in the matrix of A.

7. Placing information into a matrix
Example
Mrs smith and Mrs Jones go shopping for oranges, Apples and Bananas. Mrs Smith buys 6 oranges, 2 Apples and 4 Bananas, whilst Mrs Jones buys 2 oranges, 5 apples and 2 Bananas.
Place this information into a matrix A. 6 2 4 2 5 2

8. Multiplication of matrices
Example
Using the previous problem if we are told the price of an orange at ASDA is 34p, the cost of an Apple is 20p and the cost of a Banana is 22p, work out how much each person spent. 34 34 6 2 4 20 20 2 5 2 22 22
332
Mrs Smith spent £3.32
Mrs Jones Spent £2.12
212

9. Example
Furthermore if we look at the price of each item at Sainsbury’s the price of an orange is 44p, an apple is 40p and a Banana is 23p work out how much each would spend at this shop.
332
436
Mrs Smith spent £3.32 at ASDA £4.36 at Sainsbury’s
Mrs Jones Spent £2.12 at ASDA £3.34 at Sainsbury’s
212
334

10. In general
A 2 by 2 matrix multiplied by a 2 by 1 gives a 2 by 1 matrix
A 2 by 2 matrix multiplied by a 2 by 2 gives a 2 by 2 matrix
Example
Find the product of the following:
a) b) c)

11. Example
If and find
(i)
(ii)
NB {with multiplication of matrices order matters}

12. Matrices (1) Express as a single matrix each of the following
Express as a column vector
(i)
(ii)
(iii)
(i)
(ii)

13. x = y = p = q = a = b = 4. Find x and y given that
5. Find p and q given that
6. Find a and b when
x =
y =
p =
q =
a =
b =

14. 7. Given M, find M2 when M = 8.
(i) Show that AB = AC
(ii) Does A(B – C) = 0  A = 0 or B = C?
M2 =