Matrix Operations: Addition, Subtraction, Multiplication, Determinants, Inverses, Solving Equations, Geometric Transform

1. Unit 39Matrices

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 If what 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 Consider and First we calculate AB. ? ? ? ? ? ? ? ? ? ? ? ?

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

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 If what 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 form when You can solve for X by multiplying by This gives or 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

22. ? ? ? ?

23. ? ? ?

24. ?

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 Xʹ Yʹ Zʹ by a transformation • Calculate the coordinates of the vertices of triangle Xʹ Yʹ Zʹ • Solution ? i.e. ? ? i.e. ? ? i.e. ?

27. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is mapped onto triangle Xʹ Yʹ Zʹ by a transformation • A matrix maps triangle Xʹ Yʹ Zʹ onto triangle • Xʹʹ Yʹʹ Zʹʹ. Determine the 2 x 2 matrix, Q, which maps triangle XYZ onto Xʹʹ Yʹʹ Zʹʹ. • Solution • Xʹʹ = NXʹ = NMX so Xʹʹ = QX where ? ? ? ?

28. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is mapped onto triangle Xʹ Yʹ Zʹ by a transformation • Show that the matrix which maps triangle Xʹʹ Yʹʹ Zʹʹ back onto XYZ is equal to Q. • Solution • so QXʹʹ = X and similarly QYʹʹ = Y and QZʹʹ = Z • Thus Q maps Xʹʹ Yʹʹ Zʹʹ back to XYZ ? ? ? ? ? ? ? ? ? ? ? ? ?