1.Introduction to Matrix Algebra

7/28/2019 1.Introduction to Matrix Algebra

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I INTRODUCTION TO MATRIX A L GEB RA

Introduction to Matrix Algebra

INTRODUCTION TO MATRICESReference : Croft, A., & Davison, R. (2008). Mathematics for

Engineers - A Modern Interactive Approach, Pearson

Education.

A matrix is a rectangular array or block of numbers usually

enclosed in brackets.

A m x n matrix has m rows and n columns.Page 1


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If the matrix A has m rows and n columns we can write:

where aij represents the number or element in the ith row and

jth column.

=

mnmm

n

n

aaa

aaaaaa

A

21

22221

11211

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Special Matrices

A square matrixhas the same number of rows as columns.

The main diagonalof a square matrix is the diagonal

running from top left to bottom right.

An identity matrix, denoted by I , is a square matrix with

ones on the main diagonal and zeros elsewhere.

The transpose of A is obtained by writing rows as columns

and columns as rows, and is denoted AT.

Page 3

=100010

001

I


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Equality of Matrices

If A = (aij) and B = (bij), A = B if and only if aij= bij.

Addition and Subtraction of Matrices

Matrices of the same size may be added to and subtracted

from one another. To do this, the corresponding elements

are added or subtracted.

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e.g. 1 If

findA + B, B + C and B - C.

A + B is not defined asA andB are of not the same size.

B + C =

B C =

=

=

=

5

9

6

1

3

7

,

24

12

53

,203

412CBA

Page 5

=

+

75

85

1110

5

9

6

1

3

7

24

12

53


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Multiplication of a Matrix by a Number

Any matrix can be multiplied by a number. To do this, each

element of the matrix is multiplied by that number.

e.g.2 If , find 2A, -A.

2A =

-A =

=

8114

289

5137

A

Page 6

=

16228

41618102614

8*211*24*2

2*28*29*25*213*27*2


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Multiplication of Matrices

If A is a n x m matrix and B is a p x q matrix. For the product

AB to exist we must have m = p.

BAqpmn

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Note that matrix multiplication is :

i. not commutative (i.e. AB BA).

ii. associative [i.e. ABC = (AB)C = A(BC)].

iii. If C = AB, the element cij is found from row i of A andcolumn j of B, as follows:

c a bij ik kjk

n

==

1

ifm=p Cqn

=

ifm pdoes not exist


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=

=

2829

3626

3337

32

13

21

554

628

674

AB

Introduction to Matrix Algebra Page 8

262*63*21*8i.e.

3

1

1221=++==

=kkkbac

33 3 2 3 2


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Note that when a square matrix is post- or pre-multipliedby an identity matrix of the appropriate size the matrix is

unchanged, i.e.

AI = IA = A

Page 9

A B=

=

2 1 4

3 0 2

3 5

2 1

4 2

e.g. 3 If & , findAB.

AB=


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DETERMINANTS, INVERSE OF A MATRIX

Reference : Croft & Davison, Chapter 12, Blocks 3,4

Determinant

All square matrices, A, possess a determinant denoted by :

det(A), |A|.

Determinant of a 2 x 2 matrix

=

dc

baAIf , then det(A) = |A| = = ad - bc

dc

ba

A matrix which has a zero determinant is called singular.

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Minors and Cofactors of a 3 x 3 Matrix

Let aij be an element of a matrix A.

The minorof aijis the determinantformed by crossing out the ith

row and jth column of det(A).

The cofactorof aij= (-1)i+j x (minor of a

ij)

Note that the term (-1)i+j is called the place sign of the element

on the ith row and jth column. The following may help you to

memorize this.

++

+++

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Determinant of a 3 x 3 Matrix

Consider a general 3 x 3 matrix, A =

det(A) can be calculated by expanding along any row or

column. For example, expanding along the first row:

333231

232221

131211

aaa

aaa

aaa

|A| = a11*(its cofactor) + a12*(its cofactor) + a13*(its cofactor)

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e.g.1 Find the value of and

241

111

312

15316

52411

1741

145*33*1)2(*2

41

11*3

21

11*1

24

11*2

241

111

312

=++=

+

+

=

=

1531652411

1741


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Alternatively, by Rule of Sarrus

Repeat the 1st and 2nd column to right hand side of 3rdcolumn to form a 3 x 5 matrix.

det(A) =Add the product ofSOLID diagonals from left top to

right bottom and subtract the products ofDASH diagonalsfrom left bottom to right top.

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14

)2(*1*14*1*21*)1(*3

4*1*31*1*1)2(*)1(*2

4124111111

12312

241111

312

=

++=

=

Hence


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Properties of Determinants

i. If every element of a given row (or column) of the square

matrix is multiplied by the same factor, the value of thedeterminant is multiplied by that factor

ii. If |B| is obtained by interchanged any 2 rows (or columns) of

|A|, then |B| = -|A|.

iii. Adding or subtracting a multiple of one row (or column) to

another row (or column) leaves the determinant

unchanged.

iv. If A and B are 2 square matrices and that AB exists, then

det(AB) = det(A)det(B).

v. If 2 rows or 2 columns of a square matrix are equal, thedeterminant of the matrix is zero.

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Inverse of a Matrix

The inverse matrix of a square matrix A, usually denoted by A-1,

has the property :AA-1=A-1A = I

Note that if |A| = 0, A does not have an inverse.|A| 0, A has an inverse

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Finding the Inverse of a Matrix

The followings are steps to find the inverse of a matrix A when

|A| 0,

i. Find the transpose of A, denoted AT.

ii. Replace each element of AT by its cofactor. The resulting

matrix is called the adjoint of A, denoted adj(A).

iii.

A

AadjA

)(1 =

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e.g. 2 Find the inverse of

det(A) =14

=

241

111

312

A

=

=

=

375

173

4142

14

1

375

1734142

11

12

11

32

11

31

41

12

21

32

24

31

41

11

21

11

24

11

)(

1A

Aadj

T


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e.g. 3 Find the inverse of .

361

125

013

=B

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1.Introduction to Matrix Algebra