biz - quatitative.managment.method chapter.02

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Chapter I IChapter I I

M atrix and DeterminantsM atrix and Determinants(Part I)(Part I)

22

Matrices as Arrays:A system of m linear equations in n variables (x 1,x2,……,xn) can be arranged into a format givenbelow:

a 11x1+a 12x2+…+a 1nxn = d 1

a 21x1+a 22x2+…+a 2nxn = d 2..........................................a m1x1+a m2x2+…+a mnxn = dm



33

hree types of ingredients found in the aboveequation system:

• The first is the set of coefficients a ij.

• The second is the set of variables x 1,……..,xn.

• The last is the set of constant terms d 1,……,dm.

44

a11

a12

… a1n

a21 a22 … a 2n

…………………….

am1 am2 … a mn

A =

x1

x2

...

xn

x =

d1

d2

…

dm

d =

• Each of the above three arrays constitutes a

matrix.• A matrix is defined as a rectangular array of

numbers, parameters, or variables.

• he members of the array are called as theelements of the matrix.



55

• he array in matrix A can be written as A =[a ij] (i = 1, 2,……,m and j = 1, 2,…,n)

• A matrix (A) containing m rows and n columnsis said to be of dimension m x n (m by n).

• When m = n, the matrix is called a squarematrix.

• A matrix containing one column (such as x andd) is called column vector.

• he dimension of x is n x 1, and that of d is mx 1.

66

• If the variables x j is arranged in a horizontalarray, there would result a 1 x n matrix, whichis called a row vector.

• A row vector is distinguished from a columnvector by the use of the primed symbol:

x = [x 1 x2 … xn]

• A vector (whether row or column) is anordered n-tuple, and it may be interpreted as apoint in an n-dimensional space.

• he m x n matrix A can be interpreted as anordered set of m row vectors or as an orderedset of n column vectors.



77

Matrix Operations:

• Two matrices A = [a ij] and B = [b ij] are said to

be equal if and only if they have the samedimension and have identical elements in thecorresponding locations in the array.

Addition and Subtraction of Matrices:

• Two matrices are conformable for addition if they have the same dimension.

• The subtraction operation A – B may beconsidered as an addition operation involving amatrix A and another matrix (-1)B.

88

Example:

A firm maintains sales records for four productsover the past three months in two matrices withdimension 4 X 3. One matrix is for the easternregion’s stores and the other is for stores in thewestern region. The sales department wants tocompile a combined matrix. The followingdemonstrates addition of the two matrices:



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Sales in Thousands

Product 1 2 3 Product 1 2 3 Product 1 2 3

A

B

C

D

8 4 7

5 4 10

3 10 13

6 11 16

+

6 8 9

7 12 5

14 10 9

3 2 5

=

14 12 16

12 16 15

17 20 22

9 13 21

A

B

C

D

A

B

C

D

Month Month Month

East West Total

1010

Class Assignment 1:

Perform the following operations as indicated:

3 4 1

4 6 2

0 -3 0

0 1 7+

1.

-2 -1 0

-3 -4 5

-3 -1 0

2 4 -3-

2.



1111

Scalar Multiplication:• To multiply a matrix by a number – or in

matrix- algebra terminology, by a scalar – is tomultiply every element of that matrix by thegiven scalar.

Class Assignment 2:

2 -1

0 6

7 0

8 -43 +

1.

6 -4

10 12

2 4

-2 3- 30.5

2.

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Multiplication of Matrices:

• The conformability condition for multiplication of two matrices A and B is that the columndimension of A (the “lead” matrix in theexpression AB) must be equal to the rowdimension of B (the “lag” matrix).

• If A is of dimension m X n and B is of dimensionp x q, the matrix product will be defined if andonly if n = p.

• The product matrix AB will have the dimensionm x q – the same number of rows as the leadmatrix A and the same number of columns asthe la matrix B.



1313

Class Assignment 3:

Multiply the following matrices:

4 10 2

6

3

-4

a.3 1

6 4

8 0 -2

-4 2 3 b.

6 4 2

7 0 -3

4 0

3 6

-2 -7

c. 4 2 9 3

6 1 5 6

2 3 2 -3

0 4 2 -3

1 1 1 0

d.

5 0 00 5 0

0 0 5

1 3 4 -1-2 0 6 1

-3 -4 5 9

e.

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Class Assignment 4:A consumer products firm sells its popular detergent atfour prices corresponding to different sizes: $ 1.20, $1.50, $ 1.80, and $ 2.00. Sales of boxes (in thousands)per week for each size for each of four national regionsare listed below.

RegionRegionSizeSizeII IIII IIIIII IVIV

$ 1.20$ 1.20 22 0.50.5 22 0.60.6$ 1.50$ 1.50 1.51.5 11 2.22.2 1.11.1$ 1.80$ 1.80 33 22 11 33$ 2.00$ 2.00 11 1.21.2 1.51.5 1.21.2

Use matrix multiplication to find the total weekly dollar salesfor each re ion.



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Inner Product of Vectors:

• Given two vectors u and v with n elements

each, say, (u 1, u 2,….,u n) and (v 1, v2,….,vn),arranged either as two rows or as two columnsor as one row and one column, their inner product, written as

u . v = u 1v1 + u 2v2 +…+ u nvn

• The inner product of two vectors is a scalar.

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• If we prepare after a shopping trip a vector of quantities purchased of n goods and a vector of their prices (listed in the correspondingorder), then their inner product will give thetotal purchase cost.

• The inner product concept is exempted fromthe conformability condition, since thearrangement of the two vectors in rows andcolumns is immaterial.



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Vector Operations:

Multiplication of Vectors

• An m x 1 column vector u, and a 1 x n rowvector v , yield a product matrix uv of dimension mxn.

Example:3

2Given u = and v = [1 4 5]

Find uv

1818

Example: Given u = [3 4] and

v = 9 , we have u v = [3(9) + 4(7)] = [55]7

• u v is a matrix, despite the fact that only a singleelement is present.

• 1 x 1 matrices behave exactly like scalars withrespect to addition and multiplication.

• In the above example, we can write u v = 55. Such aproduct is called a scalar product.

• While a 1 x 1 matrix can be treated as a scalar, ascalar cannot be replaced by a 1 x 1 matrix at will if further calculation is to be carried out, unlessconformability conditions are fulfilled.



2121

b1

b2

b3

b4

b5

b6

B =

• The total amount earned on investments for thesix banks (a single dollar amount) is determinedby multiplying matrix A and matrix B.

• AB = a 1b1 + a 2b2 + a 3b3 + a 4b4 + a 5b5 + a 6b6

• The product AB (a scalar) is a dollar figurerepresenting bank earnings.

2222

Geometric Interpretation of Vector Operations:

• A column or row vector with n elements (n-vector)can be viewed as an n-tuple, and hence as a point inan n-dimensional space (n-space).

• In Fig. a, a point (3, 2) is plotted in a 2-space and islabeled u. This is the geometric counterpart of thevector u or u = [3 2].

• If an arrow (a directed-line segment) is drawn fromthe point of origin (0, 0) to the point u, it willspecify the unique straight route by which to reachthe destination point u from the point of origin.



2323

• Since a unique arrow exists for each point, wecan regard the vector u as graphicallyrepresented either by the point (3, 2), or by thecorresponding arrow.

• Such an arrow, with a definite length and adirection, is called a radius vector.

2424

1

2

3

4

x2 (6,4)2 u

(3,2)

u

0 1 2 3 4 5 6

x1



2525

(-3,-2)

-u - 2

-1

-1-2-3 0

1

2

x2

(3,2)u

x1

(b)

26261 4 5 x 12 3

1

2

3

4

5

6

0

(1,4)

v

(4,6)(v+u)

(3,2)u



2727

v-u

(-2,2)

x2

(1,4)

1 2 x1

4

3

2

1

-1-2-3-4

-1

-2

-u

(-3,-2)

v

2828

Commutative, Associative, andCommutative, Associative, andDistributive Laws:Distributive Laws:Matrix Addition:Matrix Addition:

Matrix addition is commutative as well asMatrix addition is commutative as well asassociative.associative.

Commutative LawCommutative Law : A + B = B + A: A + B = B + A

Associative LawAssociative Law : (A+B) + C = A + (B + C): (A+B) + C = A + (B + C)Matrix Multiplication:Matrix Multiplication:

Matrix multiplication is not commutative,Matrix multiplication is not commutative,that is, ABthat is, AB ≠≠ BABA



2929

Associative Law: (AB)C = A(BC) = ABC

Distributive Law:

A(B + C) = AB + AC [premultiplication by A](B + C)A = BA + CA [postmultiplication by A]

Identity Matrix:

• Identity matrix is defined as a square matrix with 1sin its principal diagonal and 0s everywhere else.

• It is denoted by the symbol I, or I n, in which thesubscript n serves to indicate its raw (as well ascolumn) dimension.

3030

• Identity matrix plays a role similar to that of thenumber 1 in scalar algebra. For any number a, wehave 1(a) = a(1) = a. Similarly, for any matrix A, wehave

IA = AI = A

• If A is of dimension 2 x 3, premultiplication and postmultiplication of A by I would call for identitymatrices of different dimensions, namely, I 2 and I 3,respectively.

• But in case A is n x n, then the same identity matrix I ncan be used. In other words, I n A = AI n, which is anexception to the rule that matrix multiplication is notcommutative.



3131

• The special nature of identity matrix is that it makes possible, during the multiplication process, to insertor delete an identity matrix without affecting the

matrix product.For example, A I B = (AI)B = A B

(mxn) (nxn) (nxp) (mxn) (nxp)

• If A = I n, then we have AI n = (I n)2 = I n

• An identity matrix squared is equal to itself. Thegeneralization is that (I n)k = I n (k = 1, 2,…..)

• Any matrix with such a property (namely, AA = A )is referred to as an idempotent matrix.

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Null Matrix:

• Just as an identity matrix I plays the role of thenumber 1, a null matrix – or zero matrix – denoted by0, plays the role of the number 0.

• A null matrix is simply a matrix whose elements areall zero.

• Unlike I , the zero matrix is not restricted to square.• A square null matrix is idempotent, but a nonsquare

one is not.

• Null matrices obey the following rules of operation(subject to conformability) with regard to additionand multi lication.:



3333

A + 0 = 0 + A = A(mxn) (mxn) (mxn) (mxn) (mxn)

A 0 = 0 and 0 A = 0

(mxn) (nxp) (mxp) (qxm) (mxn) (qxn)Idiosyncracies of Matrix Algebra:

• In the case of scalars, the equation ab = 0 alwaysimplies that either a or b is zero, but this not so inmatrix multiplication.

AB =2 4

1 2-2 41 -2

= 0 00 0

= 0

Although neither A nor B is itself a zero matrix.

3434

• For scalars, the equation cd = ce (with c ≠ 0) impliesthat d = e. The same does not hold for matrices.

C =2 3

6 9D = 1 1

1 2E = -2 1

3 2

We find that CD = CE =5 8

15 24

Even though D ≠ E.

• Here A, B, C are special matrices known assingular matrices. These matrices contain a rowwhich is a multiple of another row.



3535

Transposes:

• When the rows and columns of a matrix A are

interchanged – so that its first row becomesthe first column, and vice versa – we obtainthe transpose of A, which is denoted by A’ or

A T.

• Thus a row vector x’ constitutes the transposeof the column vector x.

• If a matrix A is m X n, then its transpose A’ must be n X m.

• An n X n square matrix possesses a transposewith the same dimension.

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• Any square matrix D is said to be symmetricmatrix if D = D’.

D =1 0 4

0 3 7

4 7 2

= D’

• he identity matrix is a symmetric matrix

because I’ = I.Properties of Transposes:

1. (A’)’ = A

2. (A + B)’ = A’ + B’

3. AB ’ = B’A’



3737

Class Assignment 5:

For the following vector X and matrix A

X = 4 2 1 5

2 0 2 -5

3 6 4 2

1 3 0 2

1 -1 -3 6

A =

Find: a. A ′ b. XA c. AX ′ d. XA ′ e. XX ′

f. AA′

3838

Class Assignment 6:A firm has employment records in two arrays. Oneincludes the number of worker-hours used during thepast month in each of five job classifications at threeproduction facilities. These data are:

Number of Worker Number of Worker --Hours at ProductionHours at ProductionFacilityFacility

JobJobClassificationClassification

11 22 3311 40004000 30003000 2000200022 30003000 30003000 4000400033 20002000 45004500 5000500044 20002000 25002500 1000100055 30003000 10001000 25002500



3939

Another array includes the average hourly wage for each job classification at each production facility as follows:

Average Hourly Wage at ProductionAverage Hourly Wage at Production

FacilityFacilityJobJob

ClassificationClassification11 22 33

11 $6.00$6.00 $5.00$5.00 $5.50$5.50

22 5.005.00 5.005.00 6.006.00

33 7.007.00 8.008.00 9.009.00

44 8.008.00 9.009.00 8.508.50

55 8.008.00 10.0010.00 9.009.00

4040

Use matrix multiplication to find:

a. Total wages paid during the month at eachproduction facility.

b. Total wages paid during the month for each jobclassification.

Include in each answer matrix only thoseelements relevant to answering the problemand leave blank all other matrix positions.

biz - quatitative.managment.method chapter.02

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