4thCOMALGE

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COMALGE (College Algebra for Business and Economics)

COMALGE (College Algebra for Business andEconomics)

Kristine Joy E. Carpio

Department of Mathematics

De La Salle University – Manila

Term 1 2011-2012

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Outline

Systems of Linear Equations

Introduction to Matrix AlgebraBasic Matrix Operations

Solving Linear Equations Using MatricesGaussian Elimination Method

Cramer’s RuleMatrix Inversion Method

References

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Course Description

This is a 3-unit course in College Algebra, specifically designed

for Business and Economics students to provide them with asolid and working knowledge of pre-Calculus Algebra. Thecourse tackles the real number system, polynomials, algebraicfractions and radicals, different methods of solving, systems of equations and their respective applications to business and

economic situations.

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DefinitionAny collection of two or more equations is called a system of

equations.The ordered pair of numbers (p, q ) is called a solution of thelinear equations

Ax + By = C

Dx + Ey = F

if its coordinates satisfy each equation.

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Methods of Solution

1. Graphical Method

2. Substitution Method

3. Elimination Method

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Graphical Solution

The graph of a linear equation is a line and points that satisfyboth equations lie on both lines. For some systems these pointscan be found by graphing. If the lines intersect at a single pointthen the equations are independent or the system isindependent. If the two lines are parallel then there is nosolution and the equations are inconsistent or the system is

inconsistent. If the two equations of the system are equivalentthen the equations are dependent or the system is dependent.

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Exercises

Determine if the system is consistent, inconsistent or dependentand solve the consistent system.

1. x − 2y = 3

3x − 6y = 8

2.

3x + y = 1

2x + 4y = −6

3. 3x − 2y = 9

6x − 4y = 16

4.

x − 2y = −1

3x + 4y = 17

Solve graphically to the nearest half-unit.

1.

5x + 2y = 10

2x + y = 42.

x + y = 1

4x + 3y = 2

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Substitution MethodFor substitution we replace a variable in one equation with anequivalent expression obtained from the other equation. Ourintention in this subsitution step is to eliminate a variable andto give us an equation involving only one variable.

1. Solve one of the equations for one variable in terms of theother. Choose the equation that is easiest to solve for x

and y .

2. Substitute into the other equation to get an equation in

one variable.3. Solve for the remaining variable (if possible).

4. Insert the value just found into one of the originalequations to find the value of the other variable.

5. Check the two variables in both equations.

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Elimination Method

1. Write both equations in the same form (usuallyAx + By = C ).

2. Multiply one or both of the equations by appropriate

numbers (if necessary) so that one of the variables will beeliminated by addition or subtraction.

3. Add/Subtract the equations to get an equation in onevariable.

4. Solve for the remaining variable (if possible).

5. Substitute the value obtained for one variable into one of the original equations to obtain the value of the othervariable.

6. Check the two variables in both equations.

COMALGE (C ll Al b f B d E )

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Exercises

Solve by substitution and elimination.

1.

3x + y = 7

2x + 3y = 7

2.

2x + 3y = 0

x + 2y = 1

3.

3

r −

1

t = −3

2

r +

7

t = 17

4.

4

r −

3

t = 1

5

r +

2

t = −4

5.

3x + 2y + 2z = −15x − 3y + 4z = −3

2x + y + 2z = −2

6.

3x + 2y − 5z = 1−13y − 24z = 11x + 5y + 8z = −5

COMALGE (C ll Al b f B i d E i )

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Introduction to Matrix Algebra

Matrix

DefinitionA rectangular array of real numbers with m rows and n columns

a11 a12 · · · a1 j · · · a1n

a21 a22 · · · a2 j · · · a2n

......

. . ....

. . ....

ai 1 ai 2 · · · aij · · · ain

... ... . . . ... . . . ...am 1 am 2 · · · amj · · · amn

is called an m × n matrix.


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Properties of Matrices

1. If m = n then it is called a square matrix.

2. An m × n matrix consists of m rows (horizontal) and n

columns (vertical).3. Each aij is called an element of the matrix. The entry aij is

the element in the i th row and j th column.

4. The order or dimension of a matrix is m × n .

5. A matrix with m = 1 has one row and is called a rowmatrix. A matrix with n = 1 has one column and is calleda column matrix.


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Properties of Matrices

6. Two m × n matrices are equal if and only if each element of one is equal to the corresponding element of the other.

7. The matrix obtained by interchanging the rows andcolumns of a matrix A is called the transpose of A, and itis written as AT . If A is m × n , then AT is n × m .


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Exercises

Find the valued of a, b, c and d .

1.

1 a 3c 4 −2

=

b −5 d

0 4 −2

2.−2 a

−1 0b 7

= c 3

−1 06 d

State the order and find the transpose.

1.

2 1 53 −6 2

2.

−1 4 04 5 20 2 3

3.

3 45 8

4.

3 −21 50 4


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Basic Matrix Operations


1. Scalar MultiplicationThe product of a number k and the matrix A is the matrixobtained by multiplying each element of A by k .

2. Addition

The sum of two m × n matrices A and B is the m × n

matrix obtained by adding the corresponding elements of A and B.

If each element of a matrix is 0, then the matrix is called a zero

matrix and it is written as 0. It is the identity for matrixaddition if that addition is defined since

A+ 0 = A,

for any matrix A.


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( g g )




3. MultiplicationThe product of two matrices A and B is written AB, andit is defined if and only if the number of columns in A isequal to the numbers of rows in B. In general, the product

of an m × n matrix A and n × p matrix B is defined, and itis an m × p matrix AB. The element cij in the i th row and j th column of the product AB is

cij = ai 1b1 j + ai 2b2 j + · · · + ain bnj .

Multiply a row in the first matrix by a column in thesecond matrix, element by element, and add products.

Note:

1. Matrix multiplication is not commutative.

2. Distributive Law does not hold.


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( g g )



Exercises

Let A =

3 −1 24 0 5

and B =

−2 3 01 5 −4

. Evaluate:

1. 2A

2. BT +A

3. −B

4. −3A

5. AT +B

6. 4B−A

7. 3A+ 2B

8. 2A− 5B

Find the product of the following matrices.

1.

2 1 −33 0 1

−2 2 5

20

1

2.

1 −24 33 5

32

3. 1 2 4

3 −1 0

1 32 −1

4 0

4.

4 −21 30 5

2 −13 0


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Remarks

If A is m × n and AT is its transpose, then

1. A ·AT is a square matrix of order m .

2. AT ·A is a square matrix of order n .

If A is an n × n matrix and if each entry in the main diagonalis 1 (the diagonal from upper left to lower right) and all theother elements are zero, the matrix is called an identity matrix

since it acts as identity for matrix multiplication. It isdesignated by I or In .


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Solving Linear Equations Using Matrices

Gaussian Elimination Method

Suppose we have the following system of n linear equations in

the n variables x 1,

x 2, . . . ,

x n

a11x 1 + a12x 2 + · · · + a1n x n = k 1

a21x 1 + a22x 2 + · · · + a2n x n = k 2...

an 1x 1 + an 2x 2 + · · · + ann x n = k n

Then each of the following operations will produce systems

which are equivalent to the given system.

1. Interchange two equations.

2. Multiply any equation a nonzero constant.

3. Add to a multiple of any equation to another equation.


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Definition (Coefficient Matrix, Augmented Matrix)

The matrix whose elements are the coefficients of the givensystem of equations, in the same relative position, is called thecoefficient matrix. If the constant terms are included on theright of the coefficient matrix as another column, the new

matrix is called the augmented matrix.For the augmented matrix of a system of linear equations,applying the elementary row operations below will produce thematrix of an equivalent system of linear equations.

1. Interchange any two rows of the augmented matrix.2. Multiply any row of the augmented matrix by a nonzero

constant.

3. Add a nonzero multiple of any row to any other row, termby term.


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The elementary row operations will be abbreviated as

Ri ↔ R j k · Ri Ri + k · R j

The method involves applying elementary row operations tochange the augmented matrix into one which represents anequivalent set of equations, but which is easier to solve. Thegoal is to produce a new augmented matrix in which:

Only 0 occurs below the main diagonal Only 1 or 0 occurs on the main diagonal, 1 if possible

This produces a matrix which allows the corresponding systemof equations to be solved by substituting known values backinto the previous equations.


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Exercises

Solve each system using the Gauss-Jordan elimination method.

1. x + y = −6

3y = 6

2.

x − y = −1

2x − y = 2

3.

6x − 7y = 02x + y = 20

4. 2x − 3y = 1

−6x + 9y = −3

5.

2x + y + z = 4x + y − z = 1

x − y + 2z = 2

6.

−x + z = −22x − y = 5y + 3z = 9


S

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Cramer’s Rule

Determinant of a 2 × 2 Matrix

The determinant of the matrix A = a b

c d is defined to be the

real number ad − bc. We writea b

c d

= ad − bc.


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Cramer’s Rule

Cramer’s Rule (2 × 2)

The solution to the system

a1x + b1y = c1

a2x + b2y = c2

is given by x =D x

D and y =

D y

D , where

D =a1 b1a2 b2

, D x =c1 b1c2 b2

, and D y =a1 c1a2 c2

,

provided that D = 0.


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Cramer’s Rule

Cofactor

Definition (Minor)

The minor of an element aij is the determinant of the matrix of order n − 1 that remains after deleting the i th row and j thcolumn, and it is written as M ij .

The cofactor of the element aij is

Aij = (−1)i + j M ij .



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Cramer’s Rule

Determinant of an n × n Matrix

The determinant of a square matrix of order n can be expressedin terms of the cofactors of the i th row as

|A| = ai 1Ai 1 + ai 2Ai 2 + · · · + ain Ain .

Similarly, it can also be expressed in terms of the cofactors of the elements of the j th column as

|A| = a1 j A1 j + a2 j A2 j + · · · + ain Anj .

In short, to find the value of a determinant:1. Choose any row or column.

2. Multiply each element in that row or column by itscofactor.

3. Add these products.



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Cramer’s Rule

Cramer’s Rule (3 × 3)

If D = 0, the solution of the system of three equations in threevariables is

x =D x

D y =

D y

D and z =

D z

D ,

where D is the determinant of the coefficient matrix, D x is thedeterminant of the matrix found by replacing the coefficients of

x in the coefficient matrix by the constant terms, and D y andD z are found similarly.



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Cramer’s Rule

Exercises

Solve each system using Cramer’s Rule.

1. 2x + 3y = 3

3x + 2y = 7

2.

5x + 3y = 74x + 5y = 3

3.

x + 5y = 33x − 2y = 9

4. 4x − 5y = 7

3x + 4y = −18

5.

2x − 3y − 2z = 103x − 4y + 3z = 84x − 5y + 4z = 10

6.

6x + 5y + 4z = 87x − 5y + 3z = 265x − 2y − 6z = −9



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Matrix Inversion Method

If A is an n × n matrix and a matrix B exists such that

AB = I = BA,

then B is called the inverse of A, and we write B = A−1.



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g q g


Inverse of a 2 × 2 Matrix

If A = a b

c d and |A| = 0, then the inverse of A is

A−1 =

1

|A|

d −b

−c a

.

Note: An n × n matrix A has an inverse if and only if |A| = 0.



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g g


Inverse of an n × n Matrix

If A is a square matrix with |A| = 0 and Aij is the cofactor of aij , then

A−1 =

1

|A|

A11 A12 · · · A1 j · · · A1n

A21 A22 · · · A2 j · · · A2n

......

. . ....

. . ....

Ai 1 Ai 2 · · · Aij · · · Ain

... ... . . . ... . . . ...An 1 An 2 · · · Anj · · · Ann

T

.



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A system of linear equations can be written in the form

AX = B,

where A is the coefficient matrix and X and B are the columnmatrices with n elements consisting of the variables and theconstant terms.

The solution of the matrix equation AX = B is X = A−1B,where A is a square matrix of order n with |A| = 0.



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Exercises

Solve each system using the Matrix Inversion Method.

1.

3x + 2y = 72x + 5y = 12

2.

3x + y = 112x + 3y = 7

3.

x + 2y = 5

3x + 2y = 11

4.

3x + 2y = 122x + 3y = 13

5.

2x + y = 3

3x + 5y = 1

6.

2x + 3y = 0

x + 2y = 1

7.

2x + y = −13x + 2y = 0

8.

x + y + 2z = 73x + 4y + 6z = 21

2x + 3y + 5z = 16

9.

2x + 2y + 3z = 93x + 3y + 4z = 13

x + 2y + 4z = 9


References

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References

1. M.J. Acelajado, Y.B. Beronque and F.F. Co. Algebra:

Concepts and Processes. National Bookstore, ThirdEdition, 2005.

2. M. Dugopolski. Algebra for College Students. McGraw-HillPublishing Company, Fourth Edition, 2006.

3. G. Fuller. College Algebra. Litton Educational Publishing,Inc., Fourth Edition, 1977.

4. P.K. Rees, F.W. Sparks and C.S. Rees. College Algebra.McGraw-Hill Publishing Company, Tenth Edition, 1990.

5. E.P. Vance. Modern College Algebra. Addison-WesleyPublishing Company, Inc., Third Edition, 1975.

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4thCOMALGE

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