lesson 1 linear algebra

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LESSON 1

LINEAR ALGEBRA

Matrix Algebra

What is matrix?

In this lecture, we will lear n about the tools for

solving linear systems of equations

There are several methods we have lear ned beforesuch as solving by graphical method,by using

substitution and elimination method

and etc.

Content

Matrix

Determinant

± Diagonal Method ± Cofactors Method

Elementary Row Operations

verse Matrix

¹¹¹

©©©

mnm2m1

2n2221

1n1211

What is Matrix A rectangular array of real (or complex)

numbers arranged in m rows and n columns

Dimension

of the matrix

Column

Some Special MatricesSquare matrix

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1.70.4

2.61.2D

¹¹¹

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2105.0

¹¹¹

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Identity matrix, nI

¹¹ º

¸©©ª

Basic oper ations of matrix

(+, -, x, T)

Matrix Addition

A is a m x n matrix and B is a m x n matrix

C = A + B

where for all i=1,2,«,m ; j=1,2,«,nijijij bac !

¹¹¹

©©©

333231

232221

131211

¹¹¹

©©©

333231

232221

131211

¹¹¹

©©©

!¹¹¹

©©©

333332323131

232322222121

131312121111

333231

232221

131211

bababa

Example

Notice that matrices with different dimension cannot

be added together:

No valid result for A + B

¹¹ º

¸©©ª

¹¹ º

¸©©ª

113 A ¹¹

¸©©ª

¹¹ º ¸©©

º ¸©©

6432100

116123 B A

¹¹ º

¸©©ª

Matrix Subtr action

A is a m x n matrix and B is a m x n matrix

C = A - B

where for all i=1,2,«,m ; j=1,2,«,nijijij bac !

¹¹¹

©©©

333231

232221

131211

¹¹¹

©©©

333231

232221

131211

¹¹¹

©©©

!¹¹¹

©©©

333332323131

232322222121

131312121111

333231

232221

131211

bababa

Example

Notice that matrices with different dimension cannot

be subtracted together:

No valid result for A - B

¹¹ º

¸©©ª

¹¹ º

¸©©ª

113 A ¹¹

¸©©ª

¹¹ º ¸©©

º ¸©©

6432100

116123

)( B A

¹¹ º

¸©©ª

Scalar Multiplication

A is a m x n matrix and E is a scalar (real number)

C = E A

where for all i=1,2,«,m ; j=1,2,«,n

e.g. C = A

ijij ac vE!

¹¹¹

©©©

vEvEvE

!¹¹¹

©©©

333231

232221

131211

333231

232221

131211

¹¹¹

©©©

!¹¹¹

©©©

901080107010

601050104010

301020101010

C ¹¹¹

©©©

908070

605040

302010

Matrix Multiplication

A is a m x n matrix and B is a n x p matrix

C = A X B ,where C is a m x p matrix

for all i=1,2,«,m ; j=1,2,«,p

To find the element cij (i-th row and j-th column of C = AB),we have to multiply each element in the i-th row of A by thecorresponding element in the j-th column of B and addthem together.

k jik n jin ji jiij babababac1

2211 .

Matrix Multiplication

C11 from 1st row of A and 1st column of B

[a11 a12 a13«a1m] 1st row of A

[b11 b21 b31«bm1] 1st column of B

C11= a11 b11 + a12 b21 + a13 b31 +«+ a1m bm1

Example

A is a 2 x 3 matrix and B is a 3 x 3 matrix

C = A X B, expect C is a 2 x 3 matrix

Find determine C, elements by elements

C11= a11 b11 + a12 b21 + a13 b31

= 1x1 + 2x0 + 3x(-1)

¹¹ º

¸©©ª

¹¹¹

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B ¹¹ º

¸©©ª

232221

131211

ccc ABC

Example

C21= a21 b11 + a22 b21 + a23 b31 = 3x1 + 2x0 + 1x(-1)

C12= a11 b12 + a12 b22 + a13 b32 = 1x0 + 2x2 + 3x3

¹¹ º

¸©©ª

¹¹¹

©©©

B ¹¹ º

¸©©ª

232221

131211

ccc ABC

¹¹ º

©©ª

!231120

B ¹¹ º

©©ª

!! 232221

131211

Example

C22= a21 b12 + a22 b22 + a23 b32 = 3x0 + 2x2 + 1x3

C13= a11 b13 + a12 b23 + a13 b33 = 1x(-2) + 2x1 + 3x2

¹¹ º

¸©©ª

¹¹¹

©©©

B ¹¹ º

¸©©ª

232221

131211

ccc ABC

¹¹ º

©©ª

!231120

B ¹¹ º

©©ª

!! 232221

131211

Example

C23= a21 b13 + a22 b23 + a23 b33 = 3x(-2) + 2x1 + 1x2

Plug in the computed elements back to C, we have

¹¹ º

¸©©ª

¹¹¹

©©©

B ¹¹ º

¸©©ª

232221

131211

ccc ABC

¹¹ º

¸©©ª

6132 ABC

Example

What is AxB?

What is BxA?

¹¹ º

¸©©ª

¨!¹¹

¸©©ª

01001100

00001000 AB

¹¹ º

¸©©ª

00 A ¹¹

¸©©ª

¹¹ º

¸©©ª

¨!¹¹

¸©©ª

10010001

10000000 BA

A B B A v{vMultiplicatio

norder is importa

Tr anspose

The transpose of a m x n matrix A, denoted

by = B, is the n x m matrix where

jiij ab !

¹¹ º

¸©©ª

¨! 654

DeterminantsIf A is square matrix then the determinant

function is denoted by det and det(A) is defined

to be the sum of all the signed elementary

matrices of A.

nnn2n1

2n2221

1n1211

The result of a

determinant isa singlenumber.

!! AAdet

Finding Determinants

l Meth

od´Determinant function for a 2×2 matrix.

aaaa!! AAdet

2211aa!

1221aa

Example 1

Compute the det(A) for the following matrix.

¹¹ º

©©ª

53Adet ! 26! 206 !

l Meth

od´Determinant function for a 3×3 matrix.

332211aaa!

132231aaa

333231

232221

131211

312312aaa

322113aaa

112332aaa

122133aaa

Example 2

¹¹¹

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Example 2«solution

8 0 8- 0!B 20

Minor & Cof actor If A is a square matrix then the minor of ,

denoted by , is the determinant of the

submatrix that results from removing the i th row

and j th column of A.

ji M1C

If A is a square matrix then the cof actor of ,

denoted by , is the number

Example 3

For the following matrix compute the minor, M23

and the cofactor, C23

¹¹¹

©©©

Example 3«solution

!740259

Cofactor

³Meth

od of Cof a

ctors´

nnn2n1

2n2221

1n1211

If A is an n×n matrix.

Choose any row, then:

nini2i2i1i1i Ca.....CaCa !A

Choose any column, then:

jn jn j2 j2 j1 j1 Ca.....CaCa !A

cof actor

Example 4

¹¹¹

©©©

Example 4«solution

2!1142

04262 !

Choose first row, then:

312111 C0C2C2 !A 312111 M0M-2M2 !

¹¹¹

©©©

Elementary Row Oper ations

Elementary row operations are manipulations

that can be performed on the row of a matrix.

¹¹ º

¸©©ª

Three basic elementary operations.

1. Interchange any two rows.1. Interchange any two rows.

¹¹ º

¸©©ª

4321 R R !

Elementary Row Oper ations

¹¹ º ¸©©

2. Multiply any row by a non2. Multiply any row by a non--zerozero

scalar.scalar.

¹¹ º ¸©©

434211

R 2R !

¹¹ º

¸©©ª

3. Add k times of any row to another 3. Add k times of any row to another

row.row.

¹¹ º

¸©©ª

85211 R R 2R !

Example 5

Use elementary row operation to transform

¹¹ º

¸©©ª

31into ¹¹

¸©©ª

¸©©

Example 5«solution

¹¹ º

¸©©ª

122 2R R R! ¹¹ º

©©ª

10062:2R

¹¹ º

¸©©ª

10R R 22

211 3R R R !

Inverse Matrix

If A is square matrix and we can find

another matrix of the same size B, such

Then we call A invertible and we say that

B is an inverse of the matrix A

nIABBA !!

Remarks on Inverse Matrix

Inverse matrix is only for square matrix

The inverse for a matrix is unique

The inverse of a matrix A is denoted as A-1

If A is invertible, then A is a non-singular

matrix

If A is not invertible, then A is a singular matrix

¹¹ º

©©ª

Finding Inverse Matrix

³Using Row Oper a

tion´Inverse for a 2×2 matrix.

operation

r owelementary

Find row operations that will convert the

first 2 columns into I2.

The third and fourth columns should then

contain A-1.

¹¹ º

©©ª

¨22A v 2I

Example 6

Determine the inverse of matrix A.

¹¹ º

¸©©ª

Example 6«solution

Form a new matrix

31Matrix

122 2R R R! ¹¹

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¨ 0131

12-100

0262:2R

1042:R

! ¹¹ º

¸©©ª

¨ 0131

110 101

Example 6«solution

¹¹ º

¸©©ª

¹¹ º

¸©©ª

211 3R R R !01

0131:R

The inverse of matrix A is

¹¹ º

©©ª

Finding Inverse Matrix³Using Row Oper ation´

Inverse for a 3×3 matrix.

operation

r owelementary

Find row operations that will convert the

first 3 columns into I3.

The last three columns should then contain

¹¹ º

©©ª

¨33A v 3I

Finding Inverse Matrix³Using Row Oper ation´

Please refer the attachment

Finding Inverse Matrix³Using Adjoint Cof actor´

If matrix

is invertible, its inverse will be

¹¹ º ¸©©

dc baA

¹¹ º

¸©©ª

Example 7

Determine the inverse for the following

matrix.

¹¹ º

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¹¹ º ¸©©

ª¨v!1

1¹¹ º ¸©©

Finding Inverse Matrix³Using Adjoint Cof actor´

If matrix

vertible, its inverse will be

¹¹¹

©©©

333231

232221

131211

Aad jointA

For Example please refer the attachment

lesson 1 linear algebra

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