Matrix and Determinants-33

8/11/2019 Matrix and Determinants-33

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Determinants

a square array of numbers enclosed by two bars

and is subjected to mathematical operation. The

elements of which have corresponding numbers

of rows to that of the columns.

where:

aij= the element of the ith row and jth column

333231

232221

131211

aaa

aaa

aaa

D


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Sign of Operators

D


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Properties of Determinants:

1. If the value of a single row or column are all 0

then D = 0.

2. If two rows or columns are interchanged, the

sign of the determinant is changed

0630520

410

D

963

741

852

963

852

741

D


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3. If each element of a row or column of a

determinant can be multiplied by a common

factor then the determinant is multiplied by that

number.

common factor =4

4

645

534

412

6165

5124

442

D


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4. If two rows or columns are identical then D = 0.

identical

0

633

522

411

D


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5. If two rows or columns are proportional then D is

equivalent to 0.

proportional

0

963

842

721

D


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6. If the corresponding rows and columns of adeterminant are interchanged, its value isunchanged.

987

654321

963

852741

D


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7. If three determinants D1, D2and D3have

corresponding equal elements except for a

single row or column in which the elements at

D1are the sum of the corresponding elementsof D2and D3then D1= D2+ D3

321

333231

232221

131211

333231

232221

131211

33323131

23222121

13121111

DDD

aab

aab

aab

aaa

aaa

aaa

aaba

aaba

aaba


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9. The value of a determinant is the algebraic

sum of the products obtained by multiplying

each element of a column or row by its co-

factor or signed minor.(Expansion of Determinants by Minor)


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Ex. Expansion by Row

D = a11 a12 a13

a21 a22 a23

a31 a32 a33

=(+) a11 a22 a23 + (-) a12 a21 a23+ (+) a13 a21 a22

a32 a33 a31 a33 a31 a32


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Minor and Cofactors

The Minor of the element Aijin the ith row and jthcolumn in any determinant order formed fromthe element remained after isolating the ith rowand jth column.

M13= a21 a22a31 a32

M23= a11 a12

a31 a32where :

Mij= the minor of Aij


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Minor and Cofactors

The cofactor of the element Aij in anydeterminant of order n is that signed minordetermined by,

Dij= ( -1 )i + j ( Mij)

D13 = ( -1 )1+3( M13)

= ( +1 ) a21 a22

a31 a32D23 = ( -1 )

2+3( M23)

= ( -1 ) a11 a12

a31 a32

where :

Dij = the cofactor of Aij


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Example

Find the cofactor using the minor of the given

matrix

D = 1 2 3

-2 3 1

3 2 1


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EVALUATION OF DETERMINANTS

1. Conventional Method

used for 2nd degree determinantsandcommonly denoted as cross product method.

Ex. Find the determinant

D = 2 1

8 5


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Ex. Find its determinant using diagonal/basket

method.

D = 2 5 4

5 0 1

1 -3 3


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3. Expansion by Minor Cofactor Method

used for 3rd degree and higher degree order

of determinants.

a. Expansion by Row (Laplaces Expansion)

n

k

ikikDAD

1


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b. Expansion by Column

n

k

kjkjDAD1


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Find the determinants of the given matrix using

expansion by column.

D = 1 4 3

4 4 5

2 5 4


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Chios Method

Another method in evaluating the

determinant of an (m x m) order matrix where

a11is not equal to zero.


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Chios Method (3x3)

333331

1311

3231

1211

2321

1311

2221

1211

)2(

11

33333231

232221

131211

)(

1.det

aa

aa

aa

aa

aa

aa

aa

aa

aA

aaaaaa

aaa

A

m

where: m is the size of

the square matrix


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Chios Method (4x4)

where: m is the size of

the square matrix

444441

1411

4341

1311

4241

1211

3431

1411

3331

1311

3231

1211

2421

1411

2321

1311

2221

1211

)2(

11

4444434241

34333231

24232221

14131211

)(

1.det

aa

aa

aa

aa

aa

aa

aa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aa

aA

aaaa

aaaa

aaaa

aaaa

A

m


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Chios method.

D = 1 4 3

4 4 5

2 5 4


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Techniques in Altering the

Elements of the Determinants.

used for 3rd degree and higher degree orderof determinants.

a. Alteration by zeroThe element of any row (or column) may bemultiplied by a constant and the result addedto the corresponding element of any other row

(or column) without changing the value of thedeterminants.


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D = A B C

D E F

G H I

D = A B C

0 E F

0 0 I

D = ( A x E x I )


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alteration by zero.

D = 1 4 3

4 4 5

2 5 4


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Elements of the Determinants.b. Pivotal Element Method

Steps:

1. Select a pivot element except zero.

2. Draw cancellation lines along the row andcolumn of the pivotal element.

3. Replace the remaining element bysubtracting from the original element, theproduct of the elements intersecting the

cancellation lines and perpendicular linesdividing it by the pivot element.

4. Multiply the resulting determinant by thepivot element along with its corresponding sign

of cofactor.


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pivot element method.

D = 1 4 3

4 4 5

2 5 4


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Seatwork

1. If , what is the cofactor of

the second row, third column element?

750

421132

B


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Seatwork

2. Given ,find its determinants using

the following methods:

a. Diagonal Method

b. Expansion by row and column

c. Chios Method

d. Alteration by zero

e. Pivotal Method

335

437

626

B


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Matrix

It is a rectangular array of numbers or functionsenclosed in a pair of brackets and subject tocertain rules of operation.

A = a11 a12 a13

a21 a22 a23a31 a32 a33 3x3

A = /aij/mxnwhere: aij= element of matrix A

mxn = size of order of matrix

m = number of row matrix

n = number of column matrix

Note: m & n may or may not be equal


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Special Type of Matrices

1. Row Vector Matrix

A matrix which contains only one row and

several columns.

Ex.

B = [ 1 2 3 4 .. n ] 1 x n


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2. Column Vector MatrixA matrix which contains only one column andseveral rows.

Ex.C = 1

2

3

.

.

.

n m x1


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3. Square Matrix

It is a matrix whose elements have equal

number of rows and columns.

Ex.

A = 1 4 3

4 4 5

2 5 4


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4. Null or Zero Matrix

It is a square matrix whose elements are all

zeros.

Ex.

A = 0 0

0 0


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5. Diagonal Matrix

It is a square matrix wherein the values lie in

the main diagonal and the rest are all zeros.

Ex.

A = 3 0 0

0 4 0

0 0 3


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6. Unity or Identity Matrix

It is a square matrix whose elements in the

main diagonal are all 1s and the rest are all

zeros.

Ex.

A = 1 0 0

0 1 0

0 0 1


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7. Symmetric Matrix

It is a square matrix whose element Aijis equalto the element Ajior the elements of the rowscorresponds to that of the column.

Ex.

A = 1 -5 6-5 7 2

6 2 3


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8. Skew Matrix

It is a square matrix whose element Aijis equalto the negative of the element Aji.

Ex.

A = 1 5 -6

-5 7 2

6 -2 3


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9. Singular Matrix

It is a square matrix whose determinant value

is equivalent to 0.

Ex.

A = 1 4

2 8


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10. Non-singular Matrix

It is a square matrix whose determinant value

is not equivalent to 0.

Ex.

A = 1 4 3 determinant value

4 4 5 /A/ = 3

2 5 4


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MATRIX LAWS

6. For Amxn, ImA = Ain = A (I is the identity formatrix multiplication)

7. (A + B)C = AC + BC (Right distributive

law)

8. A(B + C) = AB + AC (Left distributive law)

9. A(BC) = (AB) C (Matrix multiplication

is associative)


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MATRIX OPERATION

In matrix operation, only addition, subtraction,

and multiplication are defined. Division is done

by a different technique.

1. Addition / Subtraction

-two matrices may be conformable to

addition/subtraction, if and only if the size of

the two matrices are equal.

Amxn+ Bmxn= Cmxn


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MATRIX OPERATION

Example:

Evaluate A & B

A = 1 2 3

3 2 1 2x3B = 1 2

4 1

0 1 3x2

A + B = not possible because they are not of thesame order.


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MATRIX OPERATION

Example:

Find the sum and difference of the two matricesbased on the following conditions:

a. C = A + Bb. C = BA

A = 1 4 7 B = 2 5 1

2 5 8 1 6 4

3 6 9 3x3 0 3 7 3x3


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MATRIX OPERATION

2. Multiplication

a. By scalar

Example:

A x 5 where A = 1 0 13 -4 3

4 5 2

A x 5 = 5 0 5

15 -20 15

20 25 10


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MATRIX OPERATION

b. By another matrix

-two matrices can be multiplied if the number of

column (left hand) of the first matrix is equal to

the number of row (right hand) of the secondmatrix.


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MATRIX OPERATION

A * B = a11 a12 a13 b11 b12

a21 a22 a23 b21 b22

a31 a32 a33 3x3 b31 b32 3x2


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MATRIX OPERATION

A * B = (a11b11+a12b21+a13b31) (a11b12+a12b22+a13b32)

C 11 C 12

(a21b11+a22b21+a23b31) (a21b12+a22b22+a23b32)

C 21 C 22

(a31b11+a32b21+a33b31) (a31b12+a32b22+a33b32)

C 31 C 32


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MATRIX OPERATION

3. Division

Inverse of a matrix

B = 1 = A-1

A

where: A-1inverse of matrix A


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MATRIX OPERATION

a. Inverse of a matrix

A-1 = 1 adj AT

/A/

where: /A/ = determinant value of a matrix

AT= transpose of a matrix

adj = adjoint of a matrix

Note: Division operation is not conformable tomatrices of unequal rows & columns. Hence,

this operation is restricted to square matrices

only.


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MATRIX OPERATION

Transpose of a matrix ( AT )

-to determine the transpose of a matrix,

interchange the corresponding rows and

columns of the given determinant.Example: Find the transpose of the given matrix

A = 1 4 34 4 5

2 5 4


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MATRIX OPERATION

Adjoint of a Matrix ( adj. )

-to obtain the adjoint of any matrix, replace each

element by its corresponding co-factor.

Example: Find the adjoint of the given matrix.

A = 1 4 34 4 5

2 5 4


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MATRIX OPERATION

Ex. Solve for the inverse matrix.

A = 1 4 3

4 4 5

2 5 4


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Solutions to Linear Equations

Cramers Rule

Inverse Matrices

Gauss-Jordan Methods

Matrix Decomposition


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Cramers Rule

It is a theorem, which gives an expression for

the solution of a system of linear equations with

as many equations as unknowns, valid in those

cases where there is a unique solution. The solution is expressed in terms of the

determinants of the (square) coefficient matrix

and of matrices obtained from it by replacing

one column by the vector of right hand sides ofthe equations.


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Cramers Rule

Example: Use Cramers Rule to solve the system.

4x - y + z = -5

2x + 2y + 3z = 10

5x2y + 6z = 1


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Inverse Matrices

The solution is given from the product of

the adjoint of the transpose of a matrix and

its constant.


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Inverse Matrices

Example : Use Inverse Matrices to solve the

system.

4x - y + z = -52x + 2y + 3z = 10

5x2y + 6z = 1


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GAUSS-JORDAN METHODS

Gauss-Jordan Elimination

Gauss-Jordan Reduction


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To obtain the values of the unknown variables

in a given linear equation: plot the constants

along with the coefficients of the unknown

variables and then apply alteration by zeroproducing simplified equations to solve for the

unknown variables.


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A1X1 + B1X2 + C1 X3 = D1A2X1 + B2X2 + C2 X3 = D2A3X1 + B3X2 + C3 X3 = D3

A1 B1 C1 : D1A2 B2 C2 : D2A3 B3 C3 : D3

A1 B1 C1 : D1 1

0 B2 C2 : D2 2

0 0 C3 : D3 3

X1 X2 X3 K


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To obtain the values of the unknown variables

in a given linear equation: plot the constants

along with the coefficients of the unknown

variables and then apply row-by-rowtransformation to change the given matrix to a

unity matrix thus, altering the value of the

constants yielding the values of the unknown

variables.


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A1X1 + B1X2 + C1X3 = D1A2X1 + B2X2 + C2X3 = D2A3X1 + B3X2 + C3X3 = D3

A1 B1 C1 : D1A2 B2 C2 : D2A3 B3 C3 : D3

1 0 0 : X10 1 0 : X2

0 0 1 : X3


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GAUSS-JORDAN METHODS

Example: Use Gauss-Jordan Reduction andGauss-Jordan Elimination to solve the given

system:

4x - y + z = -5

2x + 2y + 3z = 10

5x2y + 6z = 1


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Also known as LU decomposition

(Crouts/Choleskys method)


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333231

232221

131211

aaa

aaa

aaa

A

33333231

2221

11

0

00

LLL

LL

L

L

33

23

1312

10010

1

U

UU

U


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Example : Use Matrix Decomposition to solvethe system.

4x - y + z = -52x + 2y + 3z = 10

5x2y + 6z = 1


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Eigenvalues and Eigenvectors

of a Matrix


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0

0

0

0

0

0

x xM

y y

x xM

y y

x x

M Iy y

xM I

y

M is a matrix and is a

scalar constant

Rearranging

In order to factorise

scalar must turn into a

matrix by multiplying it bythe identity matrix.

now it can be factorise



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00

0

0

0

0

xM Iy

a bM

c d

a bM I

c d

a bM I

c d

a b x

c d y

If the determinant of (M-I) was non-zero, it could

be inverse and multiplied

by the RHS.

Write (M- I) in the

following way and then

simplify the equation.

Write M as a matrix



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1

0

0

0

0

00

a b x

c d y

x a b

y c d

xy

If the determinant of (M-I) was non-zero, it could

be inverse and multiply it

by the RHS.

This would mean that the

vector was zero.

This means that thedeterminant of (M- I)

must be zero so is

singular.



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2

0

0

0

a b

c d

a d bc

a d ad bc

This is called the characteristic equations and

will allow to find the eigenvalues(characteristicvalues)



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x xM

y y

Once the Eigenvalues have found, substitute

these back to find their corresponding

Eigenvectors.

Eigenvectors represent Invariant Lines.

These are the lines of points that map onto

themselves after a transformation.

This represents the

Eigenvector. It is not

unique as anymultiple of it would

still be an

Eigenvector!

Ei l d Ei t


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Find the eigenvalues and correspondingeigenvectors of matrix A.

32

41

A

S t k


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Seatworks

1. Find the value of x, y, and z using InverseMatrix and Matrix Decomposition.

x + y + z = 122x + 5y -3z = 6

3x + 3y +3z = 36

2. Find the eigenvalue of24

63C

H k


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Homework

1. Given the matrices A, B, C and D

Determine:

a. BD b. A-Cc. 2A+C d. A/B

745834

763

A 35

23

12

B

132446

387

C

724

656 D


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Matrix and Determinants-33

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