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    +    = + + + +     =     (b) 3 0 2 4 1 4 1 1 1 2 3 0 3 1 0 1 2 1 4 2 1 3 4 0 − −    −    = − − − − − − − −     = − − −     2 1 3 2 4 4

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-3 Example 1 If A = 4 6 5 7 3 1 9 4     and B = 2 8 3 1 5 2 4 6 − −     , determine (a) A + B; (b) B + A and (c) A - B. Thus matrix addition is commutative. ( ie A + B = B + A ) Example 2 Let A = , B = 4 7 5 2 and C = 5 6 8 12 6 1 2 3−             . Show that ( A + B ) + C = A + ( B + C ) Thus matrix addition is associative. [ ie ( A + B ) + C = A + ( B + C ) ]

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-4 8.5 MULTIPLICATION OF MATRICES (a) Scalar multiplication To multiply a matrix by a single number (ie a scalar), each individual element of the matrix is multiplied by that factor. eg 4 3 2 5 6 1 7 12 8 20 24 4 28x =         ie In general [ ] [ ]K Ka aij ij= (b) Multiplication of two matrices Two matrices A and B can be multiplied together if and only if the number of columns in A is the same as the number of rows in B. Let [ ] [ ]A = ; and B = a bij m x n ij n x q Then the product AB is an m x q matrix. Example 3 Find the product of 2 3 5 6 7 1 0 7−               . Example 4 Find the product of [ ]5 6 7 8 5 6 8 10 −            .

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-5 Example 5 Find the product of 2 0 0 4 3 0 5 6 1 5 6 7                     . Example 6 If A = 2 3 4 5 and B = 1 2 -1 - 3 , find AB and B A .        Example 7 If A = 1 2 3 4 and B = 1 2 3 0 1 2 3 2 1 , find A B .              AB is not defined since the number of columns of A does not agree with the number of rows of B.

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-6 8.6 PROPERTIES OF MATRIX MULTIPLICATION 1 ( K A ) B = A ( K B ) = K ( A B ) 2 A ( B C ) = ( A B ) C - Associative 3 A B ≠ B A - Not Commutative 4 ( A + B ) C = A C + B C - Is Distributive wrt addition 5 C ( A + B ) = C A + C B 6 If A B = 0 , It does not necessarily imply that A = 0 or B = 0 . 7 If A B = A C It does not necessarily imply that B = C . TUTORIAL 8 1 If A = 6 2 -1 5 and B = - 4 - 5 - 6 - 7         , find: (a) A + B; (b) B + A and (c) A - B 2 If A = 1 2 3 -1 - 2 - 3 4 5 6 and B = 3 -1 2 0 3 -1 0 1 3                     , find: (a) A + B and (b) 2 A - 2 B . 3 If A = 3 - 5 1 - 2 B = 2 0 3 1 and C = 1 2 3 4 ,           , (a) verify the associative law ( A + B ) + C = A + ( B + C ); (b) find 3 A - B - C 4 If A = 1 3 2 4 , B = 0 1 2 3 and I = 1 0 0 1             , find ( I - 2 A + 4 B ).

MATRICES Mathematics Preparatory Course for Direct Entry to Higher Nitec 8-7 5 Determine the values of the variables for which the following main equations are true :