Chpt 8 - Matrices

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-1 CHAPTER 8 8.1 DEFINITION: A matrix is a set of real or complex numbers (or elements) arranged in rows and columns to form a rectangular array. a a a a a a a a a a a a n n m m m mn 11 12 13 1 21 22 23 2 1 2 3 .......... .......... . . . .......... . . . . .......... . ..........                 The numbers a11 , a12 , a13 , ............ amn are called the elements of the matrix. The order of a matrix is specified according to the number of rows and number of columns it possesses. A matrix with m rows (the horizontal lines) and n columns (the vertical lines) is called an (m x n) matrix or the matrix is said to be of order m x n . Double - subscript notation is used for the elements in the matrix. The first subscript denotes the row and the second subscript the column containing the given element. eg a i j is the element in the i th row and j th column. Matrices may be denoted by capital bold-faced letters. eg A , B , [ ]aij , [ ]aij mxn 8.2 EQUALITY OF MATRIX Two matrices are said to be equal if: i ) they are the same order, ii ) their corresponding elements are equal So, if a a a a a a 11 12 13 21 22 23 4 6 5 2 3 7    =     then a a a a11 12 13 214 6 5 2= = = =; ; ; ; etc eg 1 4 9 2 5 8 1 2 3 1 1 2 1 3 1 2 2 2 2 2 2    = + + +         5 3 4 7 13 1 x y x y + −    =     means 5 3 13 4 7 1 x y x y + = − =

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-2 8.3 TYPES OF MATRICES (a) Row Matrix A matrix having only one row and is of order 1 x n . eg [ ]x x x1 2 3 (b) Column Matrix A matrix having only one column and is of order m x 1 . eg y y y 1 2 3           (c) Square Matrix A matrix having the same number of rows and columns. Obviously, only one number is needed to specify its order. eg 2 3 1 4               ; 3 1 2 12 2 7 5 3 20 8.4 ARITHMETIC OF MATRICES Addition and Subtraction of Matrices Two matrices A and B of the same order can be added (or subtracted) by adding (or subtracting) their corresponding elements. For example : (a) 1 2 3 4 7 8 9 10 1 7 2 8 3 9 4 10 8 10 12 14    +    = + + + +  