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Matrix

Session 1

Course : K2092 - Linear Algebra

Year : December 2011

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DefinitionA matrixis a rectangular array of numbers. Thenumbers in the array are called the entriesin the matrix.

A general mx nmatrixAas

The entry that occurs in row iand columnj of matrixAwill be denoted aijor (A)ij.

nmmnmm

n

n

aaa

aaa

aaa

A

...

...

...

21

22221

11211

Bina Nusantara University

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Example 1 Examples of matrices

Some examples of matrices

Size (order) of matrix is denoted by m x n, withm number of rows and n number of its coloumn

Example: 3 x 2, 1 x 4, 3 x 3, 2 x 1, 1 x 1

4,31,

000

10

2

,3-012,

41

03

21

21

row matrix or row vector column matrix or

column vector

entries

4

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A matrixAwith nrows and ncolumns is called asquare matrix of order n, and the shaded entries

are said to be on the main diagonalofA.

nnaaa ,,,

2211

The preceding matrix can be written as

Matrices Notation and Terminology

ijnmij aa or

nnnn

n

n

aaa

aaa

aaa

...

...

...

21

22221

11211


5

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Definition

Two matrices are defined to be equalif theyhave the same sizeand their corresponding

entries are equal.

.andallforifonlyandifthen

size,samethehaveandIf

jibaBA

bBaA

ijij

ijij


6

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Example 2Equality of Matrices

Consider the matrices

043

012,5312,

312 CBx

A

If x= 5, thenA =B.

For all other values of x, the matricesAand Bare not

equal.

There is no value of xfor whichA= C, why??


7

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Example 3Addition and Subtraction

Consider the matrices

Then

The expressions A+C, B+C, A-C, and B-C are

undefined.

2211,

5423

1022

1534

,

0724

4201

3012

CBA

51141

5223

2526

,

5307

3221

4542

BABA

Bina Nusantara University 9

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Scalar MultiplicationIfAis any matrix and cis any scalar (a real or complexnumber), then the productcAis the matrix obtained bymultiplying each entry of the matrixAby c. The matrix cA

is said to be the scalar multipleofA.

For the matrices

1203

369,

531

720,

131

432CBA

401

123,

531

7201-,

262

8642

31 CBA

We have


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Matrix Multiplication (1/2) Matrix multiplicationA is an m r matrix and B is an r

n matrix, then the productisAB is the m n matrixwhose entries are determined as follows.

the entry in row iand columnjof AB, single out row ifrom the matrix A and columnjfrom the matrix B.Multiply the corresponding entries from the row andcolumn together and then add up the resultingproducts.


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Matrix Multiplication (2/2) To find the entry in row iand columnjof AB, single out

row ifrom the matrix A and columnjfrom the matrix B.Multiply the corresponding entries from the row and

column together and then add up the resultingproducts.


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Example 4: Multiplying Matrices (1/2) Consider the matrices

Solution Since A is a 2 3 matrix and B is a 3 4 matrix, the product

AB is a 2 4 matrix. And:


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Example 4: Multiplying Matrices (2/2)


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Example 5Determining Whether a Product Is Defined

Suppose that A ,B ,and C are matrices with the following

sizes, determining which product is defined.A B C

3 4 4 7 7 3

Solution:

AB is defined and is a 3 7 matrix; BC is defined andis a 4 3 matrix; and CA is defined and is a 7 4

matrix. The products AC ,CB ,and BA are all undefined.


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Properties of Matrix Operations For real numbers a and b ,we always have ab=ba, which

is called the commutative law for multiplication.

For matrices,AB and BA need not be equal.

Equality can fail to hold for three reasons:

The product AB is defined but BA is undefined.

AB and BA are both defined but have different sizes. It is possible to have even if both AB and BA

are defined and have the same size.

BAAB


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Example 6:AB and BA Need Not Be Equal

examples.countergivingforusefulbemaymatricetw oThese

BA?toequalisABDoes,10

10Band

00

11A


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TheoremProperties of Matrix Arithmetic

Assuming that the sizes of the matrices are such that the indicatedoperations can be performed, the following rules of matrix arithmetic arevalid:


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Example 7Associativity of Matrix Multiplication


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Cancelation Law does not hold (1)

Recall that for any real numbers a,b,c

ab= acand a 0 means b= c

This called cancelation law. However, this does not hold in

matrix multiplication.Example:Consider

It can be verified that

altough B C


43

52,

43

11,

20

10CBA

ACAB

8643

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Transpose

If A is any mn matrix, then the transpose ofA,denoted by ,is defined to be the nmmatrix

that results from interchanging the rows andcolumns of A; that is, the first column of isthe first row of A ,the second column of is thesecond row of A ,and so forth.

TATA

TA


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Example 8Some Transposes (1/2)


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Example 9Some Transposes (2/2)

Observe that

In the special case where A is a squarematrix, thetranspose of A can be obtained by interchanging entriesthat are symmetrically positioned about the main diagonal.

jiij

T AA )()(


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TraceIf A is a square matrix, then the trace of A,denoted by tr(A), isdefined to be the sum of the entries on the main diagonal of A .Thetrace of A is undefined if A is not a square matrix.


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Zero Matrices A matrix, all of whose entries are zero, such as

is called a zero matrix.

A zero matrix will be denoted by 0 ; if it is important toemphasize the size, we shall write 0mxnfor the mn zeromatrix. Moreover, in keeping with our convention of using

boldface symbolsfor matrices with one column, we willdenote a zero matrix with one column by 0 .

Zero matrix has roles as identity element of matrixmultiplication, similar with0 as identity element ofmultiplication in real numbers.


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Theorem: Properties of Zero Matrices

Assuming that the sizes of the matrices are such thatthe indicated operations can be performed ,the

following rules of matrix arithmetic are valid.


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Cancelation Law Does Not Hold (2)

Recall that for any real numbers a and b

a0= b0means a= b

However, this does not hold in matrix multiplication.Example:

It can be verified that

altough neitherAnor D is zero matrix.


00

73,

20

10DA

00

00

AD

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Identity Matrix Square matrices with 1s on the main diagonal and 0s off

the main diagonal such as is called an identity matrix.Identity matrix with size is denoted by In

Identity matrix takes role as identity element of matrixmultiplication,similar with 1 as identity elemnet ofmultiplication on real number. That is, if defined,AI =Aand IA=A.


I2 I3 I4

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Recall : A = A and A=A, asAis an mnmatrix

Example 10Multiplication by an Identity Matrix

mI

nI


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If A is a square matrix, and if a matrix B of the same size can be foundsuch that AB = BA = I , thenA is said to be invertible and B is calledan inverse of A (B=A-1). If no such matrix B can be found, then A issaid to be singular .

Invers of matrix


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TRIANGULAR MATRIX

UPPERTRIANGULAR MATRIX

A square matrix with all elements BELOW its main

diagonal are zeros

a b c

A = 0 d e

0 0 f

LOWERTRIANGULAR MATRIX

A square matrix with all elements ABOVE its main

diagonal are zerosBina Nusantara

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LOWER TRIANGULAR MATRIX

A square matrix with all elements ABOVE its main

diagonal are zeros

a 0 0

B = b c 0

d e f

Bina Nusantara

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Exercise If A =

find (AB + C)t


20

02,

01

10

01

,111

121CB

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k0292001022011402301 matrix.ppt

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