1. Unit 39 Matrices Presentation 1Matrix Additional and Subtraction Presentation 2Scalar Multiplication Presentation 3Matrix Multiplication 1 Presentation 4Matrix Multiplication 2 Presentation 5Determinants Presentation 6Inverse Matrices Presentation 7Solving Equations Presentation 8Geometrical Transformations Presentation 9Geometric Transformations: Example

2. Unit 39 39.1 Matrix Additional and Subtraction

3. If a matrix has m rows and n columns, we say that its dimensions are m x n. For example is a 2 x 2 matrix is a 2 x 3 matrix You can only add and subtract matrices with the same dimensions; you do this by adding and subtracting their corresponding elements. ? ?

4. Example 1 (a) (b) ? ? ? ? ? ? ? ? ? ?

5. Example 2 Ifwhat are the values of a, b, c and d? Solution Subtracting gives Hence ? ? ? ? ? ? ? ?

6. Unit 39 39.2 Scalar Multiplication

7. For scalar multiplication, you multiply each element of the matrix by the scalar (number) so Example If then ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

8. Unit 39 39.3 Matrix Multiplication 1

9. ? ? ? You can multiply two matrices, A and B, together and write only if the number of columns of A = number of rows of B; that is, if A has dimension m x n and B has dimension n x k, then the resulting matrix, C, has dimensions m x k. To find, C, we multiply corresponding elements of each row of A by elements of each column of B and add. The following examples show you how the calculation is done. Example If and, then A is a 2 x 2 matrix and B is a 2 x 1 matrix, so C = AB is defined and is a 2 x 1 matrix, given by: ? ? ?

10. Unit 39 39.4 Matrix Multiplication 2

11. Here we show a matrix multiplication that is not commutative Considerand First we calculate AB. ? ? ? ? ? ? ? ? ? ? ? ?

12. Is AB = BA? No Hence matrix multiplication is NOT commutative Here we consider a matrix multiplication that is not commutative Considerand And now for BA. ? ? ? ? ? ? ? ? ? ? ? ? ?

13. Unit 39 39.5 Determinants

14. ? For a 2 x 2 square matrix its determinant is the number defined by Example 1 What is detA if ? Solution ? ? ? ?

15. ? ? ? ? For a 2 x 2 square matrix its determinant is the number defined by Example 2 Ifwhat is the value of x that would make detM = 0 ? Solution ? ? A matrix, M, for which detM = 0 is called a singular matrix.

16. Unit 39 39.6 Inverse Matrices

17. For a 2 x 2 matrix, M, its inverse, is defined by You can always find the inverse of M if it is non-singular, that is. For Example If find and verify that Solution Hence ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? where

18. Unit 39 39.7 Solving Equations

19. You can write the simultaneous equation In the formwhen You can solve for X by multiplying by This givesor So we first need to find. Now and Hence ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

20. Unit 39 39.8 Geometrical Transformations

21. You can use matrices to describe transformations. We write where is transformed into Lets look at the common transformations

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25. Unit 39 39.9 Geometric Transformations: Example

26. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-1, 4) is mapped onto triangle X ʹ Y ʹ Z ʹ by a transformation (a)Calculate the coordinates of the vertices of triangle X ʹ Y ʹ Z ʹ Solution ? ? ? ? ? ? i.e.

27. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is mapped onto triangle X ʹ Y ʹ Z ʹ by a transformation ? ? ? ? (b)A matrixmaps triangle X ʹ Y ʹ Z ʹ onto triangle X ʹʹ Y ʹʹ Z ʹʹ. Determine the 2 x 2 matrix, Q, which maps triangle XYZ onto X ʹʹ Y ʹʹ Z ʹʹ. Solution X ʹʹ = NX ʹ = NMX so X ʹʹ = QX where

28. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is mapped onto triangle X ʹ Y ʹ Z ʹ by a transformation (c)Show that the matrix which maps triangle X ʹʹ Y ʹʹ Z ʹʹ back onto XYZ is equal to Q. Solution so QX ʹʹ = X and similarly QY ʹʹ = Y and QZ ʹʹ = Z Thus Q maps X ʹʹ Y ʹʹ Z ʹʹ back to XYZ ? ? ? ? ? ? ? ? ? ? ? ? ?