Matrices A matrix is a table or array of numbers arranged in rows and columns The order of a matrix is given by stating its dimensions. This is known as a matrix of order 2 × 3 since it has two rows and three columns. The element of A in the i th row and j th column is denoted a ij. For example a 12 = 4, a 21 = 3 and a 23 = 2. B is a 3 ×1 column matrixC is a 1 × 4 row matrixD is a square matrix of order 2

Addition Scalar Multiplication Subtraction

The Transpose of a matrix It is sometimes convenient to switch rows and columns. When the rows and columns of matrix A are interchanged, the resulting matrix is called the transpose of A denoted A’ or A T A matrix is symmetrical if A = A T

A matrix is Skew Symmetric if A T = -A Note there can only be zeros in the leading diagonal. Some other Rules:

Page 4 Exercise 1 Questions 1, 2, 3a, 4a, c, e, 6g, i, p, r, t, 7a, f, 9, 10 Page 7 Exercise 2 TJ Exercise 1, 2, 3 and 4

Matrix Multiplication Matrix A can only be multiplied by matrix B when the number of columns in matrix A is the same as the number of rows in matrix B. A and B might be compatible to form AB but not BA. The product of an m × n matrix with an n × p matrix will result in an m × p matrix.

Page 10 Exercise 3 Questions 1a, c, 2a, c, k, m, o, 3a, 4, 5a, c Page 11 Exercise 4A Questions 6, 7, 8 TJ Exercise 5

Summary Multiplying by the identity matrix does not change the matrix. (i.e.×1)

Page 13 Exercise 4B – as many as you can.

The Determinant of a 2×2 Matrix The augmented matrix will be Performing ERO’s we can reduce this to A solution exists only if ad – bc ≠ 0 Cayley called this number, ad – bc, the determinant of the matrix. The determinant is denoted by det(A) or |A|.

The Determinant of a 3×3 Matrix Using the same principals from the previous page on a 3×3 matrix, which you follow on pages 22 and 23, the determinant of a 3×3 matrix is; Page 16 Exercise 5 Questions 1b, d, h Page 25 Exercise 7 Questions 4, 5a, b

The inverse of a 2×2 Matrix

Hence the solution is x = 2, and y = -1.

Page 19 Exercise 6A Questions 1, 2, 4, 8, 9 (some) TJ Exercise 8

The Inverse of a 3×3 Matrix Place A and I side by side, with A on the left. Perform ERO’s with a view to reducing it to I. Perform the same ERO’s on I. When finished I will represent A -1

Page 28 Exercise 8 Question 1, 3 TJ Exercise 9, 10, 11.

Transformation Matrices In computer animation, an object may be drawn by joining lists of points, defined by their coordinates. These points are then transformed according to a rule in order to make the object move. In this section we will be studying such transformations – linear transformations. Consider that, under a transformation the point P(x, y) has an image P’(x’, y’). Then

A triangle has vertices O(0,0), A(2,5), B(4,0). Find its image under the transformation with associated matrix Hence O’ (0,0), A’ (9,10), B’ (8,0).

Constructing a Transformation Matrix To find the transformation matrix we only need consider the images of (1,0) and (0,1).

Find the matrix R associated with a reflection in the line y = -x. Calculate the coordinates of a typical point (x,y) under this transformation. (1,0)  (0,-1) (0,1)  (-1,0) a = 0, c = -1 b = -1 d = 0 Thus P’ (-y, -x)

(a) Find the matrix k associated with an anticlockwise rotation of  0 about the origin. (b) Find the coordinates of the image of P(2,4) under this transformation with  =60 0. 00 00

Page 32 Exercise 9A Questions 1, 2, 5(some), 6 TJ Exercise 12 END OF TOPIC