MATRICES AND DETERMINANTS

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INTRODUCTIONMatrix is a powerful 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 ‘ .

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MATRICES A matrices is 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 57 9 8

A =2 x 3

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TYPES OF MATRICESo Square matrixo Rectangular matrixo Row matrixo Column matrixo Diagonal matrixo Scalar matrixo Null matrixo Unit matrix or Identity matrixo Sub matrixo Triangular matrix

i. Upper triangular matrixii. Lower triangular matrix

o Transpose of a matrixo Symmetric matrixo Skew symmetric matrixo Idempotent matrix o Order of a matrixo Trace of a matrix 4

MATRIX OPERATIONSAddition and subtraction of matriceso 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 additiono Commutative lawo A + B = B = Ao Associative lawo A + ( B + C ) = ( A + B ) + Co Distributive lawo k ( A + B ) = kA + kbo Additive identityo A + 0 = A = 0 + Ao Existence of inverseo A + ( - A ) = 0 = ( - A ) + Ao Cancellation lawo A + C = B + Co => A = B

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MULTIPLICATION OF MATRICES

Multiplication of 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 12 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 =

4 310 13

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.

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DETERMINANTSo A Determinant is 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

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Minor of a matrix

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

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CONCLUSIONMatrix algebra provide a 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.

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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 .

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THANK YOU

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