Matrices are widely used in business, economics, and other fields. They allow problems to be represented with distinct finite numbers rather than infinite gradations as in calculus. Sociologists, demographers, and economists use matrices to study groups, populations, industries, and social accounting. [/SUMMARY]

1. MATRICES AND
DETERMINANTS
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2. INTRODUCTION
Matrix 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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3. 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 5
7 9 8
A =
2 x 3
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4. TYPES OF MATRICES
o Square 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

5. MATRIX OPERATIONS
Addition and subtraction 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

6. 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
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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
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7. 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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8. DETERMINANTS
o 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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9. 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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10. CONCLUSION
Matrix 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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11. 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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12. THANK YOU
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