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Matrices

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 . .

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Matrices

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  1. 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 ith row and jth column is denoted aij. For example a12 = 4, a21 = 3 and a23 = 2. B is a 3 ×1 column matrix C is a 1 × 4 row matrix D is a square matrix of order 2

  2. Addition Scalar Multiplication Subtraction

  3. 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 AT A matrix is symmetrical if A = AT

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

  5. 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

  6. 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.

  7. 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

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

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

  10. 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|.

  11. 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

  12. The inverse of a 2×2 Matrix

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

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

  15. 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

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

  17. 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

  18. 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).

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

  20. 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. a = 0, c = -1 b = -1 d = 0 (1,0)  (0,-1) (0,1)  (-1,0) Thus P’ (-y, -x)

  21. (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  =600 . 0 0

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

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