Matrices

1. Matrices

2. Given below is a record of all the types of books kept in a class library.

3. The numbers in the table can be presented in a rectangular array. The first column represents the number of books which are fiction and the first row represents the number of books in Malay, and so on.

4. A set of numbers arranged in this manner is called a matrix( plural – matrices) • The numbers within each matrix are the elements. • In the above matrix, there are 3 rows and 2 columns. • This matrix has a order of 3 by 2 or it is a 3 x 2 matrix. • In general an m x n matrix refers to m rows and n columns. Its order is m x n

5. Column matrix : If the matrix only has one singular column • Row matrix : If the matrix only has one singular row.

6. Square matrix : Is when the matrix has an equal number of rows & columns. • If every elements in a matrix is zero then its is a null matrix ( zero matrix) ; It is denoted by 0

7. Identity matrix Given that matrix A has an inverse, which is A-1, Then when matrix A is multiplied with the inverse matrix A-1 , it will give a matrix called identity matrix. A A-1 = I

8. Singular Matrix. Given that the determinant of A is 0. i.e. det A = 0, then the inverse of A does not exist. When a matrix does not have an inverse then that matrix is called the Singular matrix. For example this matrix does not have an inverse, it is a singular matrix.

9. When are 2 matrices equal?? If 2 matrices A and B • are of the same order • have corresponding elements that are equal Then, we can say that they are equal.

10. Addition & Subtractionof Matrices Given that P & Q are matrices of the same order P + Q = =

11. P - Q = = Note: If the order of P and Q are not the same, their sum and difference are not defined

12. Scalar Multiplication • Multiplication of a matrix by a real number kP = where k is a real number

13. Matrix Multiplication • Possible only if no. of columns in left matrix = no. of rows in right matrix • Multiplying m x n matrix by n x p matrix will result in an m x p matrix ie: A 3x3 matrix multiplied by a 3x2 matrix result in a 3x2 matrix. A 3x2 matrix multiplied by a 2x3 matrix result in a 3x3 matrix. A 3x2 matrix multiplied by a 3x3 matrix is not posible

14. 2 2 1 1 To multiply 2 matrices: = = 2 x 2 2 x 3 2 x 3

15. Some Properties of Matrix Multiplication • NOT commutative ie: AB ≠ BA • Is associative ie: ABC = (AB)C = A(BC)

16. Commutative and associative properties surprised to learn that matrix multiplication is not commutative but associative. Explain using a problem

17. Page 3 Commutative and associative properties Prices C A B (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.

18. General Formula for the Determinant Let A be a square matrix of order n. Write A = (aij), where aij is the entry on the row number i and the column number j, for i =1, …, n and j = 1,, n. We have for any fixed i, and for any fixed j where Aij (called the cofactors) is the determinant of the square matrix of order (n-1) obtained from A by removing the row number i and the column number j multiplied by (-1)i+j. *Determinant of n  n matrix

19. Inverse of a matrix Definition of inverse An n x n matrix A is said to be invertible if we can find an n x n matrix B such that AB = BA = I B is called the inverse of A and is usually denoted A-1.

20. Inverse of a matrix The inverse of a square matrix does not always exist. If the inverse exists, the matrix is defined to be non-singular (i.e. det A  0) If the inverse does not exist the matrix is singular(i.e. det A =0).

21. The importance of the inverse matrix can be seen from the solution of a set of algebraic linear equations such as Ax = b If the inverse A-1 exists then pre-multiplying both sides gives A-1Ax = A-1b Ix = A-1b and since pre-multiplying a column vector of length n by the nth order identity matrix I does not affect its value, this process gives an explicit solution to the set of equations x = A-1b Inverse of a matrix

22. Using matrices to solve Simultaneous Equations • We can solve simple simultaneous equations like x +3y = 6 and 2x + y = 4 by using matrices. • It can be represented by matrix equation x + 3y 6 2x + y 4 or 1 3 x 6 2 1 y 4

23. How to solve simultaneous equations using matrices? • This is in the form of AX = B where A 1 3 X x and B 6 2 1 y 4 If A has an inverse ( ie A-1) Then A-1 A X = A-1 B ie I X = A-1 B X = A-1 B

24. To find the variables x and y • First, determine if determinant A exists det A = ( 1 x 1) – (2 x 3) = 1 – 6 = -5 A-1 1 1 -3 -5 -2 1 Hence, X = A-1 B 1 1 -3 6 -5 -2 1 4 2 3 5 4

25. Thus, x 6 y 8 5 5 To check, subsitute x = 6/5 and y =8/5 back into the original set of equations.

26. Other Applications • We are given two matrices X and Y that show the price of 3 brands of cat food sold at two stores, for months May and June respectively. • X 9.4 9.6 Y 9.8 10.0 10.2 10.4 10.2 10.2 11.4 11.0 12.0 12.0

27. ai) Evaluate Y - X • First, determine if the operation can take place. This means that the order of the two matrices are the same. Y – X = 9.8 - 9.4 10.0 - 9.6 10.2 - 10.2 10.4 - 10.2 12.0 - 11.4 12.0 - 11.0 = 0.4 0.4 0 -0.2 0.6 1.0

28. Explain what the numbers in the answer represent. • The values in the matrix shows the increase or decrease in prices of the three brands of cat food by comparing the June prices against the May prices.

29. b) Write down a matrix M such that XM will show the total costs for each brand of cat food • First, decide which matrix to use. In this case, its matrix X. Write down the order of the matrix. • Thus, matrix X is a 3 x 2 matrix.

30. Know order of final matrix • Also, there is a need to know the order of the final matrix. In this case, its going to be a 3 x 1 matrix to show the total cost for each of the brands. • X M = Total cost for 9.4 9.6 brand A 10.2 10.4 brand B 11.4 11.0 brand C

31. To find the order of matrix M • X M = Total cost • ( 3 x 2) M = ( 3 x 1 ) • Thus M needs to be a (2 x 1) matrix.

32. To find total cost, multiply the two matrices together Total cost 9.4 9.6 50 In cents = 10.2 10.4 50 11.4 11.0 950 = 1030 1120

33. What are the values of m1, m2, m3? • The amount spend in May and June for Cat food on Brand A = 950 cents = $9.50 Brand B = $10.30 and Brand C = $ 11.20 • The prices are given in 50 cents per 100g. • If the Question asks for 10 tins of 500g, thus we need to multiply the cost 50 cents by 50.

34. Page 1 Matrices Storing and organising numerical data 6 rows and 7 columns of data, we say the matrix has size (order) 6  7

35. explain why the matrix of prices can be a 31 or 13 matrix in (a) and what information is found if V is post-multiplied by Page 2 Operations with Matrices 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)?

36. R = Page 4 Routes Matrices B A E the loop at B gives 2 routes from B to B but the loop at D gives only 1 route because it is one-way only. C D