Matrices. Introduction. Matrices have many important uses. One of the simplest uses of matrices is to store data. In fact, this is the most common use outside mathematics.
Matrices are a powerful unifying concept, connecting ideas in mathematics to those in other content domains as well as connecting various branches of mathematics. As such, matrices can be a means of integrated whole rather than a set of isolated topics.
6 rows and 7 columns of data, we say the matrix has size (order) 6 7
(e) 3 by 5
(f) 2 by 1
(g) 1 by 1
What do you notice about the order of the two matrices involved in each case?
( P – Q ) .
* A matrix has the same number of rows and columns.
Matrix multiplication was invented in the 19th Century as a useful way of combining two suitable matrices. It multiplies elements in a row with elements in a column to produce one single number.
To make up some matrix multiplication problems for yourself. And to investigate the condition on the sizes of two matrices that is necessary before you can be multiplied.
Example : Maybeline bought 3 apples, 2 oranges and 4 pears.The cost of each item is 25 cents, $1 and 40 cents respectively. Find the total cost of the purchase.
Answer : The total cost of the purchase
= 3 25 cents + 2 100 cents + 4 40 cents
= 435 cents
= $ 4.35
Using Matrices :
explain why the matrix of prices can be a 31 or 13 matrix in (a) and what information is found if V is post-multiplied byMatrices Multiplication
The prices of the drinks per cup are 55¢ for tea, 60¢ for coffee and 75¢ for drinking chocolate.
(a) Form a matrix of prices and use it to find the total amount taken on each of the 3 days.
(b) What information would be found by pre-multiplying V by (1 1 1)?
matrix multiplication is not commutative but associative.
Explain using a problem
(AB)C is calculated by first finding the number of doors and windows in each of the developments and then finding the total cost of windows and doors for each development.
What information does the matrix AB gives? BA?
A(BC) is calculated by first finding the cost of the doors and windows in each model and then finding their total cost for each development.
Calculate (AB)C and A(BC). Explain why they are equal.
If you combine the matrix I with any matrix P and the result is the matrix P, then I is known as the identity matrix.
Is there a 2 2 identity matrix for matrix multiplication. i.e.
AI = IA = A , where I is the identity matrix.
For example, I =
N.B. When referring to the multiplicative identity, it is usually called "the identity matrix".
What do you obtain when A is multiplied by
the identity matrix?
AI = A or IA = A
Given A and the inverse of A, denoted by A-1
IMPT NOTE :
Step 1 : Find the determinant of the matrix A, denoted by det A
Step 2 : The inverse of matrix A is
How to prove it?
Using Matrices to Solve Simultaneous Equations
the equations have no unique solution.
The interior design company is given the job of putting up the curtains for the windows, sliding doors and the living room of the entire new apartment block of the NTUC executive condominium. There are a total of 156 three-bedroom units and each unit has 5 windows, 3 sliding doors and 2 living rooms. Each window requires 6 m of fabric, each sliding door requires 14 m of fabric and each living room requires 22 m of fabric. Given that each metre of the fabric for the window cost $12.30, the fabric for the sliding door costs $14.50 per metre and each metre of the fabric for the living room is $16.50.
We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium.
The property market in Singapore went up very rapidly in the 1990’s. Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream. NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties. With the onset of the Asian economic crises, the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes. Do you think this is a good request? How long do you think it will take to set up a company to run the property business?
The Microsoft Excel matrix functions are:
of an array
matrix of an array
array. The first row of the input
becomes the first column of the
output array, etc.
Matrices can be used to store data about graphs. The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points. If the edges are assigned a direction, the graph is called directed.
Matrices are also used in cryptography, the art of writing or deciphering secret codes.
A spreadsheet can be used for the tedious matrix operations as shown below.
In this code, the word PEACE looks like 11 22 26 24 22.
Suppose we want to encode the message: MATHS IS FUN
If we decide to divide the message into pairs of letters, the message becomes MA TH IS SF UN.
Now transform the column vectors by multiplying each of them on the left by A:
The encoded message is 106 66 71 45 70 44 79 50 51 32.
To decode, multiple by A-1 and reassigning letters to the numbers.