Math unit39 matrices
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This document outlines the key concepts and examples for matrices including: addition and subtraction of matrices with the same dimensions; scalar multiplication by multiplying each element of the matrix by the scalar; matrix multiplication where the number of columns of the first matrix equals the number of rows of the second matrix; determinants of 2x2 matrices; inverse matrices for non-singular 2x2 matrices; solving systems of equations using matrices; and geometric transformations using matrices including rotation, reflection, translation and examples of applying transformations.
Math unit39 matrices
- 1. Unit 39 Matrices Presentation 1 Matrix Additional and Subtraction Presentation 2 Scalar Multiplication Presentation 3 Matrix Multiplication 1 Presentation 4 Matrix Multiplication 2 Presentation 5 Determinants Presentation 6 Inverse Matrices Presentation 7 Solving Equations Presentation 8 Geometrical Transformations Presentation 9 Geometric 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. ? ?
- 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. Is AB = BA? No Hence matrix multiplication is NOT commutative Here we consider a matrix multiplication that is not commutative Consider and And now for BA. ?? ? ? ?? ? ? ? ? ? ? ?
- 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. ou can write the simultaneous equation the form when ou can solve for X by multiplying by his gives or o we first need to find . Now and Hence ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?
- 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 X Y Zʹ ʹ ʹ by a transformation (a) Calculate the coordinates of the vertices of triangle X Y Zʹ ʹ ʹ 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 X Y Zʹ ʹ ʹ by a transformation ? ? ? ? (b) A matrix maps triangle X Y Zʹ ʹ ʹ onto triangle X Y Z .ʹʹ ʹʹ ʹʹ Determine the 2 x 2 matrix, Q, which maps triangle XYZ onto X Y Z .ʹʹ ʹʹ ʹʹ Solution X = NX = NMXʹʹ ʹ so X = QXʹʹ where
- 28. Example A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is mapped onto triangle X Y Zʹ ʹ ʹ by a transformation (c) Show that the matrix which maps triangle X Y Zʹʹ ʹʹ ʹʹ back onto XYZ is equal to Q. Solution so QX = Xʹʹ and similarly QY = Yʹʹ and QZ = Zʹʹ Thus Q maps X Y Zʹʹ ʹʹ ʹʹ back to XYZ ? ? ? ? ? ? ? ? ? ? ?? ?