Mathematics SKE: STRAND J. UNIT J4 Matrices: Introduction

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1 UNIT : Learning objectives This unit introduces the important topic of matrix algebra. After completing Unit J4 you should be able to add and subtract matrices of the same dimensions (order) understand the differences between scalar and matrix multiplication and calculate each with confidence understand the meaning of determinant, singular matrices and adjoint matrices and be able to calculate the inverse of a 2 2 square matrix, where it exists be able to solve sets of linear simultaneous equations using matrix methods be able to express and use transformations in terms of their matrix representation. A matrix is a rectangular array of numbers. The history of matrices (and determinants) originates from the use of magic squares and Latin squares. For magic squares, the array of numbers is such that all the row and column sums and the also the sums of both diagonals, are equal (to the 'magic constant', 'magic number' or 'magic sum'). Magic squares were known to Chinese mathematicians as early as 650 BC. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad from around 983 AD; simpler magic squares were known to several earlier Arab mathematicians. Some of these magic squares were later used in conjunction with magic letters to assist Arab illusionists and magicians. The magic square below is from ancient India; it is known as the Kubera-Kolam floor pattern and used the numbers 20-28, with magic constant 72. This is just one interesting use of arrays of numbers; the more significant application of matrices came with their development to solve sets of equations. The Chinese text, Nine Chapters on the Mathematical Art, written during the Han Dynasty (206 BC AD), gives the first known example of matrix methods. First a problem is set up: "There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?" Now the author does something quite remarkable. He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board' (in reverse order of the context)

2 UNIT : Our 21st century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical. Most remarkably the author, writing in 200 BC, instructs the reader to multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives Next the left most column is multiplied by 5 and then the middle column is subtracted as many times as possible. This gives from which the solution can be found for the third type of corn, then for the second, then the first by back substitution. This method, now known as Gaussian elimination, would not become well known until the early 19th century. The modern developments of matrix algebra did not really begin until the 19th century; these were based on the work of the British mathematician Arthur Cayley ( ). In 1858, Cayley published 'A Memoir on the Theory of Matrices'. In this paper, he explained how to do basic arithmetic with matrices, including addition, subtraction, scalar multiplication and matrix multiplication. Matrix addition and multiplication satisfy many (but not all) of the properties of ordinary addition and multiplication. Cayley showed how to use these properties to solve algebraic equations involving matrices. Arthur Cayley ( ) Key points and principles You can only add or subtract matrices that have the same dimensions (order), that is, the same number of rows and columns as one another For scalar multiplication, you multiply every element of the matrix by the scalar Matrix multiplication is only defined for C = AB if the number of columns of A = number of rows of B; the dimensions (order) of C are (number of rows of A) (number of columns of B) A square matrix has equal numbers of rows and columns The determinant of a square matrix M is singular if det M = 0 The inverse of a square matrix M exists if det M 0 and M 1 = adj M det M 2

3 UNIT : You can solve the set of linear simultaneous equations, AX = B provided det A 0; then X = A 1 B Reflections, rotations and enlargements can all be expressed in the form X' = MX when M represents the transformations. Facts to remember An ( m n) matrix has m rows and n columns Matrix multiplication, AB, is only defined if the number of columns of A = number of rows of B a b If M =, det M = ad cb If det M = 0, the matrix is singular adj M d = c b a M 1 adj M det M = Writing a set of linear simultaneous equations in the form AX = B means that its solution is X = A 1 B provided A 1 exists; that is det A 0. The main transformations can be expressed as X' = MX where M is given by Reflection Reflection In the line y = x In the line y In the y-axis In the x-axis = x

4 UNIT : Rotation (Centre of rotation is the origin) Rotation 90 clockwise 90 anticlockwise 180 (clockwise or anticlockwise) Enlargement (Centre of enlargement is the origin) Enlargement Scale factor n n 0 0 n Glossary of terms Matrix an array of numbers in rows and columns. For example, Dimensions of a matrix ( m n) where m is the number of rows and n the number of columns For example, ; this is a 2 3 ( ) matrix, that is, 2 rows and 3 columns. Scalar multiplication here you multiply every element of the matrix by the scalar. For example, = Matrix multiplication C = AB is only defined if the If A is n k number of columns of A = number of rows of B ( ) and B is ( k m) matrix, then C is ( n m) matrix. Square matrix this has the same number of rows and columns. Null matrix this is the matrix, of any dimensions, which has each element zero; 0 0 for example, or

5 UNIT : Identity matrix a square matrix, denoted by I, of the form or , etc. Determinant of a square matrix If M = a b, det M = ad bc. Singular matrix M is singular if M is a square matrix and det M = 0 (and non-singular if M 0). Adjoint matrix If M is a square matrix, M = a b, the adjoint matrix, is defined by M d = c b a Inverse matrix If XA = AX = I, then X is the inverse of A and is denoted by A 1 ; you can calculate A 1 by M A 1 adj = det M 5