What is a matrix Working with matrices

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What is a matrixWorking with matrices
Matrices
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Matrices

• Matrices are stores of information and are
  defined as below.

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Matrix

• Matrix - a rectangular array of numbers arranged
  into m rows and n columns

Rows, i 1, , m
Columns, j 1, , n
NB. Opposite order to coordinates
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Examples

• These are valid matrices

• These are not

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Row and Column Matrices (vectors)

• row matrix (or row vector) is a matrix with one
  row

• column vector is a matrix with only one column

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Square Matrix

• When the row and column dimensions of a matrix
  are equal (m n) then the matrix is called square

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Definitions Definitions

• Here are more examples of matrices.
• You can see that each matrix is a different size.
• The size (or order) of a matrix is given as rows
  columns.
• What is the order of each of the above matrices?

2 2
2 5
4 1
3 3
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The milkman

• A milkman delivers 3 varieties of milk to 4
  different houses a week over 2 weeks

E
Pasteurised (PA)
F
Semi-skimmed (SS)
G
Skimmed (SK)
H
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• The number of pints of each type of milk
  delivered to each house is represented by
  matrices M and N.

How many bottles of skimmed did he deliver to
house G altogether?
Could I show the total for both weeks in a single
matrix?
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• The number of pints of each type of milk
  delivered to each house is represented by
  matrices M and N.

How much more Pasturised was delivered to house H
in 2nd week?
Could I show the increase in delivery for each
type of milk to each house in a single matrix?
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Matrix Addition
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Examples - Matrix Addition

• The following matrix addition is not defined

• Only matrices of the same order can be added or
  subtracted.

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Matrix multiplying by a scalar

• You can multiply a matrix by a number as
  illustrated in the example below.
• All that has happened is each element has been
  multiplied by the number outside the matrix.
• In general for any 2 2 matrix.
• Remember that this will work for any matrix of
  any order.

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Dairy rivalry..
Two rival milk companies advertise their prices
as shown. Can we show this using a matrix called
P?
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Comapring the companies

• Can you use the information below to put
  together a cost matrix for each house using each
  company (only in first week)? What will be the
  order of your cost matrix?

C11 represents the total cost for house E if they
use company 1.
C11
(8)(35)
(4)(32)
(3)(27)
489
C12
(8)(36)
(4)(30)
(3)(27)
489
C31
(2)(35)
(7)(32)
(6)(27)
456
Now work out the values of all the elements in C.
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Matrix multiplication
M x P C
4x3
3x2
4x2
Does the order (size) of the matrix matter when
multiplying??
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Matrix multiplication

• Matrix multiplication (A x B) is only possible if
  the number of columns in A is equal to the number
  of rows in B.

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Matrix multiplication

• Work out the following
• A B
• B A
• A B
• A X B
• B X A

Do you notice anything?
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Using your Graphic Calculator

• Try using your graphic calculator to work out the
  cost matrix for week 2.

• Click on matrix and go to edit
• Press enter to edit matrix A
• Enter order of matrix and fill in cells
• Do the same for 2nd matrix B
• Click on quit
• Now click on matrix and select A and then times
  it by B