Matrices And Determinants

Matrices And Determinants

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  MATRICES AND DETERMINANTS 1
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  INTRODUCTION Matrix is apowerful toll in modern mathematics having wide applications . Sociologists use matrices to study the dominance within a group . Demographers use matrices in the study of birth and survivals , marriage and descent , class structure and mobility , etc. . Matrices are all the more useful for practical business purpose and , therefore , occupy an important place in Business Mathematics . Obviously , because business problems can be presented more easily in distinct finite number of gradation than in infinite gradation as we have in calculus . Economists now , use matrices very extensively in ‘ social accounting ‘ , ‘ input – output table ‘ and in the study of ‘ inter – industry economics ‘ . 2
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  MATRICES  A matricesis a rectangular array of numbers arranged in rows and column and are enclosed by a pair of bracket and is subjected to certain rules of presentation.  A matrix is usually denoted by a capital letter and its elements by corresponding small letters followed by two suffices. 3 4 5 7 9 8 A = 2 x 3 3
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  TYPES OF MATRICES oSquare matrix o Rectangular matrix o Row matrix o Column matrix o Diagonal matrix o Scalar matrix o Null matrix o Unit matrix or Identity matrix o Sub matrix o Triangular matrix i. Upper triangular matrix ii. Lower triangular matrix o Transpose of a matrix o Symmetric matrix o Skew symmetric matrix o Idempotent matrix o Order of a matrix o Trace of a matrix 4
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  MATRIX OPERATIONS Addition andsubtraction of matrices o Two matrices A and B are said to be conformable for addition or subtraction if they are of the same order. o Two matrices of different orders cannot be added or subtracted. Properties of addition o Commutative law o A + B = B = A o Associative law o A + ( B + C ) = ( A + B ) + C o Distributive law o k ( A + B ) = kA + kb o Additive identity o A + 0 = A = 0 + A o Existence of inverse o A + ( - A ) = 0 = ( - A ) + A o Cancellation law o A + C = B + C o => A = B 5
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  MULTIPLICATION OF MATRICES Multiplicationof two matrices  Two matrices A and B are said to be conformable for product AB , if they are of the same order. PQ = PQ = Multiplication of a matrix by a scalar  If ‘ k ’ be a scalar and A be a matrix , then the matrix obtained by multiplying every elements of A by k is said to be the scalar multiplication of A by k. It is denoted by kA 6 0 1 2 3 -1 2 4 3 0 x -1 + 1 x 4 0 x 2 + 1 x 3 2 x -1 + 3 x 4 2 x 2 + 3 x 3 For Example : P = Q = 4 3 10 13
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  PROPERTIES OF MULTIPLICATION Distributive with respect to addition A ( B + C ) = AB + BC  Associative , if conformability is assured ( AB ) C = A ( BC )  Commutative law AB =/= BA  Multiplication by a unit matrix ( I ) A x I = A = I x A  Multiplication of a matrix by itself If A is a square matrix and in that case A x A will also be a square matrix of the same order.  If A is n x m matrix and 0 is m x n then , A x 0 = 0 = 0 x A .  If AB = 0 ( null matrix ) , it is not necessary that A = 0 , or B = 0 or both A and B are zero.  If AB = AC , where A =/= 0 does not necessarily implies B = C. 7
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  DETERMINANTS o A Determinantis a compact form showing a set of numbers arranged in rows and columns , the number of rows and the number of columns being equal . The numbers in a determinant are known as the element of the determinants. A determinant is denoted by A or A or A Order of determinants oIt is the number of rows and columns of the determinants. Determinant of order 1 oLet A = a11 then A = a11 Determinant of order 2 oLet A = then A = a11a22 – a12a21 a11 a12 a21 a22 8
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  Minor of amatrix o The minor of a element aij is the a determinant of a residual matrix obtained by deleting the ‘i’th row and ‘j’th column . The minor of an element is denoted by Mij. Co – factor of a matrix o Co – factor of an element aij is defined by aij = ( -1 ) i+j x Mij . Determinant of order 3 Let , Then , a11 a12 a13 a21 a22 a23 a31 a32 a33 A = a22 a23 a32 a33 a21 a23 a31 a33 a21 a22 a31 a32 A = a11 - a12 + a13 9
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  CONCLUSION Matrix algebra providea system of operation on well ordered set of numbers . The common operations are addition , multiplication , inversion , transpose , etc . A most significant contribution of matrix algebra is its extensive use in the solution of a system of large number of simultaneous linear equations . The widely used ‘Linear Programming ‘ has its basic in matrix algebra . It is on this account , matrix algebra is defined at times as linear algebra . In the study of communication theory and in electrical engineering the ‘ net work analysis ‘ is greatly aided by the use of matrix representation. 10
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  BIBLIOGRAPHY  Business Mathematics, Sancheti D . C & Kapoor V . K , Sultan Chand & Sons , Eleventh Edition .  Fundamentals Of Business Mathematics , Potti L . R , Yamuna Publications .  Fundamentals Of Business Mathematics , Nag N . G , Kalyani Publishers . 11
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  THANK YOU 12