Unit 39 Matrices Presentation 1 Matrix Additional and

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

Unit 39 39. 1 Matrix Additional and Subtraction

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.

Example 1 (a) (b) ? ? ?

Example 2 If what are the values of a, b, c and d? Solution Subtracting gives Hence ? ? ?

Unit 39 39. 2 Scalar Multiplication

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

Unit 39 39. 3 Matrix Multiplication 1

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: ? ?

Unit 39 39. 4 Matrix Multiplication 2

Here we show a matrix multiplication that is not commutative Consider and First we calculate AB. ? ? ? ? ? ?

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

Unit 39 39. 5 Determinants

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

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

Unit 39 39. 6 Inverse Matrices

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

Unit 39 39. 7 Solving Equations

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 ? ?

Unit 39 39. 8 Geometrical Transformations

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

Unit 39 39. 9 Geometric Transformations: Example

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 (a) Calculate the coordinates of the vertices of triangle Xʹ Yʹ Zʹ Solution ? i. e. ? ? ? i. e.

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 (b) 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 ? ?

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 (c) 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