m15 l1 matrices&determinants

8/2/2019 M15 L1 Matrices&Determinants

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Is a set of equations containing two or moreunknowns having similar solution set.

Linear SystemsIs a set oflinear equations containing two or

more unknowns having similar solution set.


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Examples:1. 2.

3. 4.

943 yx

1087 zyx

0852 yx

254 zxx

3825 xyx

0348 yx

626 zyx

7 yx

03742 zyx

10 zyx


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1. Independent or consistent system

System of equations in two or more variables represented by linesintersecting at a common point.

Have finite number of solutions represented by the points of

intersection of the lines.2. Inconsistent system

System of equations in two or more variables represented by non-intersecting curves.

System with no solution3. Dependent system

System of equations in two or more variables represented by curvescoincident with one another.

System with infinite number of solutions.


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1. 2.

3. 4.

5.

8yx

15zy3x4

1yx2

2z2yx 422 zyx

62 yx

9y6x8

1)zy(3

1x

5x3y

1)2(2

1 xzy

10y2x6

2)yx2(4

1z


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Graphical Method of Solving a System of Linear EquationsThe graphical method of solving a system of linear equations is a methodthat determines the solutionin terms of the common point(s)or thepoint(s) of intersectionamong the graphs representing each of theequations in the system.

The following are the basic steps to be followed:

1. Draw the graphs associated to the equations of the system.

2. Determine the common point or the point of intersection among the graphs.3. Read the coordinates of the point giving the solution ( x , y ).


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Find the solution to each of the given system by the graphical method.

1. 2.

1. S (3, 1) 2. S (1, 2)

9y3x2 13y4x3

X Y1 Y20 3 13/4

1 7/3 5/2

2 5/3 7/4

3 1 1

EQ1

EQ26

11

2

y

3

x

4

3

3

y

4

x

EQ1EQ2

X Y1 Y2

0 11/3 9/4

1 3 3

2 7/3 15/4

3 5/3 9/2


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Analytical MethodsElimination Of a Variable by Addition/Subtraction

This is an analytical method of solving a system ofequations that eliminates a variable addition/subtraction of

multiple equations.

Elimination Of a Variable by Substitution

This is an analytical method of solving a system ofequations that eliminates a variable by replacing one of thevariables in one of the equations by an equal expressionsobtained from the other equation.


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At the end of the lesson, the student is

expected to be able to:

Define matrix

Identify different types of matrices.

Perform operations on matrices.Define determinant of matrix.

Evaluate determinant of a square matrix.


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Matrix is a rectangular array of elements

arranged inm rows andn columns, and is

enclosed by a pair of parenthesis ( ), braces [ ]

or brackets { }. The elements maybe

numbers, variables.


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Matrix can be written in the general form:

mna...m2am1a

............2n

a...22

a21

a1na...12a11a

)ij

(aA

row column

Uppercase letterLower case letter Listed elements


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2. Row Vector or Row Matrix

6402A

is a matrix with only one row andn columns

(1xn).


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3. Square Matrix

422

303

012

A

is a matrix with equal number ofrows and

columns.


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4. Symmetric Matrix

423

201

312

A

is a matrix withaij

= aji

.


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5. Diagonal Matrix

400

070

002

A

is a square matrix with elements above and

below the main diagonal is zero.


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6. Identity Matrix or Unit Matrix

100

010

001

A

is a diagonal matrix with all elements on the

main diagonal is equal to 1.


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7. Upper Triangular Matrix

800

320

236

A

is a matrix with all elements below the main

diagonal is equal to 0.


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8. Lower Triangular Matrix

834

027

006

A

is a matrix with all elements above the main

diagonal is equal to 0.


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9. Zero One Matrix

011

00

11

0

0

A

is a matrix consisting ofzeros and ones only

as entries.


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10. Transpose of a Matrix

011

0011

0

0

A

The transpose of a matrix , denoted by AT , is

obtained by interchanging the rows and

columns of the matrix.

000

101101TA


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A. Matrix Equality

ijbaij

BA

A matrix of the same dimension (mxn) if all

elements of the first matrix is equal to its

respective element on the second matrix

for all is and js


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Find the values of a, b, c, and d so thateach of the given statements would be true:a. b.

01

2

031

522

c

ba

2

7

2

1

24

47

20

01

b

d

a

c

=


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B. Matrix Addition

ijbac ijij

BAC

The sum of matrices of the same dimension

(mxn) is the sum of all elements of the same

position. for all is and js

44

92

0422

4531

02

43

42

51


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C. Scalar Multiplication

84

102

42

512

)( ijakAk

The scalar product of matrix is obtained by

multiplying a constant k to every elemnt of

the matrix

for all is and js


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Properties of Matrix Addition and Scalar Multiplication

If A, B, C and O (zero matrix) are m x n matrices and c and d are scalar

numbers, then the following hold true.

1. A + B = B + A Commutative Property of Addition

2. A + (B + C) = (A + B) + C Associative Property of Addition

3. cdA = c(dA) Associative Property of Scalar

Multiplication

4. IA = A Scalar Identity

5. c(A + B) = cA + cB Distributive Property6. (c +d) A = cA + dA Distributive Property

7. A + O = A Identity Property of Addition

Note: The difference AB = A + (-B)


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C. Matrix Multiplication or Vector Product

)()()( kxnxmxkmxn

AxBC

A vector product of matrices A and B can be

obtained if the number ofcolumns of the

multiplicand is equal to the number ofrows

of the multiplier


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p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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Let A, B, and C be matrices with sizes so thatthe given expressions are all defined, and let c

be a real number.

|1. Multiplication is not commutative.

2. A(BC) = (AB) C

3. c(AB) = (cA)B = A (cB)

4. A(B+C) = AB + AC5. (B+C) A = BA + CA


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Suppose A, B, are mxn matrices, C is an nxpmatrix, c is a real number.

|1. (A+B)T = AT + BT.

2. (AC)T = CT AT

3. (AT) T = A

4. (cA)T = cAT


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Given the following matrices, perform

the indicated operations

a. A *B -2A b. B*AT + 2( B)

97

83

52

31BA


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)det(AA

A scalar quantity associated with a square

matrix.


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12

53A

The determinant of an order 2 matrix (2 x 2 matrix) can be

obtained by taking the difference of the products of the

diagonal elements.

40

21B

|A| = det(A) = 3*1 -2*5

|A| = -7|B| = det(B) = -1*4 -0*2

|A| = -4


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12

03B

Evaluate the following determinants

1. 4.

2. 5.

3. 6.

03

12A

41

32C

18

24D

21

53E

12

510F


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For a 3x3 matrix, the determinant is:

333231

232221

131211

||aaaaaa

aaa

A

)()()(|| 312232121333213123123223332211 aaaaaaaaaaaaaaaA


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The determinant of an order 3 matrix (3 x 3 matrix) can beobtained by doing the following steps:

1. Copy the first two columns of the given matrix as the

4th and 5th columns.

2. Multiply the downward diagonal elements of theresulting matrix. Find the sum of these three products.

3. Multiply the upward diagonal elements of the matrix.

Find the sum of these three products.

4. Subtract the result of (2) by the result of (3). This is the

determinant.


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221

103

212

A

Evaluate the following determinant:

21

03

12

|A| = det(A) = (2*0*-2+-1*1*1+2*3*2)

(1*0*2+2*1*2+-2*3*-1)

|A| = 1


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The minor of a matrixmij about an elementaij is asubmatrix of A where the ith row and jth column has

been removed.

The cofactor of a Matrix about the elementaij is the

determinant of the minormij preceded by

(negative) if(i+j) is odd.


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For any square matrix of order n>=3, the determinant can beevaluated using:

a) Co-factor Expansion about a column:

for all i, for a given j

b) Co-factor Expansion about a row:

for all j, for a given i ,

where: aij is the element in the ith-jth position

Aij is the cofactor of aij.

m

i

ijijAaA1

||

n

j

ijijAaA1

||


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221

103

212

A

Evaluate the following determinant:

Column aij mij Sign Aij aij*Aij

1 3 - -(-2) 6

2 0 + +(-6) 0

3 1 - -(5) -5

TOTAL 1

Selecting row 2

21

22

22

21

21

12


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110

322

111

B

Evaluate the following determinants using both methods

1. 3.

2. 4.

204

121

211

A

111

422

301

D

414123

031

F

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