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Matrices and Matrix Operations

Linear Algebra

MATH 2010

Basic Denition and Notation for Matrices

{Ifmandnare positive integers, then anmxnmatrixis a rectangular array of numbers (entries) mrows8 >>:2 6 664a

11a12a13::: a1n

a

21a22a23::: a2n............

a m1am2am3::: amn3 7 775
|{z} ncolumns whereaijis the number corresponding to theithrow andjthcolumn.iis the row subscript and jis the column subscript. {Thesizeof the matrix ismxn. {Matrices are denoted by capital letters:A,B,C, etc. {Ifm=n, then the matrix is said to besquare. {For a square matrix,a11,a22,a33, ...,annis called themain diagonal. {tr(A) denotes thetraceofAwhich is the sum of the diagonal elements. For example, if A=2

43 5 2

01 4

3 1 23

5 then the diagonal elements are 3, -1, and 2, so tr(A) = 3 + (1) + 2 = 4 {Acolumn vectoris a matrix with only 1 column, i.e., it has sizemx1. Example: 2 41
3 23
5 {Arow vectoris a matrix with only 1 row, i.e., it has size 1xn. Example:

01 2 8

{Two matrices areequalif they are the same size and all the entries are the exact same. For example, if A=2 4 1 3 andB=a4 1b what doaandbhave to equal forA=B? Adding and Subtracting Matrices: IMPORTANT!!! In order to add/subtract matrices, matrices must be theSAMEsize. If two matrices are the same size, then to add (subtract) them, we simply add (subtract) corresponding elements. Let

A=2 1 1

11 4 andB=23 4 3 12 Then

A+B=2 1 1

11 4 +23 4
3 12

2 + 2 1 + (3) 1 + 4

1 + (3)1 + 1 4 + (2)

42 5
4 0 2 and

AB=2 1 1

11 4 23 4
3 12

22 1(3) 14

1(3)11 4(2)

0 43 22 6
Scalar Multiplication:Example, 2A. In order to do scalar multiplication, multiply all entries by the scalar. For example, using the matrixAfrom above, i.e.,

A=2 1 1

11 4 we can calculate 2Aas

2A=2(2) 2(1) 2(1)

2(1) 2(1) 2(4)

=4 2 2 22 8
Linear Combination:IfA1,A2, ...,Anare matrices of the same size andc1,c2, ...,cnare scalars, then c

1A1+c2A2+:::+cnAn

is called alinear combinationofA1,A2, ...,Anwith coecientsc1,c2, ...,cn. For example, if

A=2 1 1

11 4 ; B=23 4 3 12 ; C=0 51 1 04 then

2A3B+C= 22 1 1

11 4 323 4
3 12 +0 51 1 04

2(2) 2(1) 2(1)

2(1) 2(1) 2(4)

3(2) 3(3) 3(4)

3(3) 3(1) 3(2)

+0 51 1 04 4 2 2 22 8
69 12
9 36 +0 51 1 04

46 + 0 2(9) + 5 212 + (1)

2(9) + 123 + 0 8(6) + (4)

2 1611

85 10
Properties of Matrix Addition and Substraction:LetA,BandCbemxnmatrices andcand dbe scalars, then

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

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

3. (cd)A=c(dA) Associative Property of Scalar Multiplication

4. 1A=AMultiplicative Identity

5.c(A+B) =cA+cBDistributive Property

6. (c+d)A=cA+dADistributive Property

Transposes:

{The transpose of a matrix is denotedAT. To nd the transpose of a matrix, you interchange the rows and columns. In other words, you can think about it as write all the rows as columns or all the columns as rows. For example, if A=1 2 31
then A T=1 3 21

Find the transpose of

A=2 41 2
0 5 323
5 {Notice that ifAismxn, thenATisnxm. {Some Properties of Transposes

1. (AT)T=A

2. (A+B)T=AT+BT

3. (cA)T=cAT

{A matrix is said to besymmetricifA=AT.

Sample Problems:

1. Let

A=21 5

0 3 5

FindAT.

2. Let

A=2

41 2 3

0 5 4 32 13
5 andB=2 431
0 1 5 43 5

Find (A+ (2B)T)T.

3. Findcanddso that

A=2

41 2 5

2 3c d4 03 5 is symmetric.

Special Matrices:There are two special matrices,

{ Identity Matrixis denoted byIorInwherendenotes a square matrix of sizenxn. The identitity matrix is a square matrix with 1's on the diagonal and 0's as all other elements: I 2=1 0 0 1 ; I 4=2 6

641 0 0 0

0 1 0 0

0 0 1 0

0 0 0 13

7 75
{ Zero Matrix Thezero matrixis denoted byOorOmxnwhereOis a matrix of sizemxn. This is simply a matrix with all zeros. Example: O=0 0 0 0 ;orO=2 40 0
0 0 0 03 5

Properties of the Zero Matrix

1.A+O=Awhere it is understood thatOhas the same size asA.

2.A+ (A) =O

3. IfcA=O, thenc= 0 orA=O.

Matrix Multiplication:Matrix multiplication is more involved. You canNOTmultiply correspond- ing entries!! {To help understand the process of matrix multiplication, we will rst examine an applied problem which uses the same strategy as is used in matrix multiplication. Assume you are at a football stadium where there are three dierent refreshment centers, the south stand, north stand and west stand. At each stand, they are selling peanuts, hot dogs and soda. See the gure below. Assume you want to know how much total the south stand made. You need to multiply the number of each of the items sold by the south stand (in the rst row of the matrix) by the selling price of each item (given in the column vector containing selling price). In other words, you need to isolate the rst row and multiply by the corresponding items in the column and then add:

120 250 3052

42:00
3:00 2:753 5 = 120(2:00) + 250(3:00) + 305(2:75) = 1828:75:

So, the south stand sold a total of $ 1828.75.

Similarly, the north stand sold

207 140 4192

42:00
3:00 2:753 5 = 207(2:00) + 140(3:00) + 419(2:75) = 1986:25:

And the west stand sold

29 120 1902

42:00
3:00 2:753 5 = 29(2:00) + 120(3:00) + 190(2:75) = 940:50: What we just did was matrix multiplication. We multiplied a 3x3 matrix by a 3x1 matrix to get a 3x1 matrix: 2

4120 250 305

207 140 419

29 120 1903

52
42:00
3:00 2:753 5 =2

41828:75

1986:25

940:53

5 { Size of matrices is important!Notice above, that we multiplied two matrices together, one was size 3x3 and the other was size 3x1. They are NOT the same size. LetAbe amxnmatrix andBbe apxqmatrix. In order to multiplyAB,the number of columns ofAmust equal the

number of rows ofB. The schematic below will help.So, ifAbe amxnmatrix andBbe apxq, then in order to multiplyAB,nmust equalpand the

resulting size ofABismxq. { Examples:First, determine if it is possible to ndABandBAby looking at the sizes of the matrix. If so, what is the size of the resulting matrix? Find the resulting matrix.

1.A=1 2

2 1 ,B=32 4 2

2.A=1 2

3 4 ,B=01 2 3

3 4 0 1

3.A=2 42 1
3 4 1 63 5 ,B=2 401 0
4 0 2 81 73
5 { Properties of Matrix Multiplication:LetA,B, andCbe matrices of appropriate size for matrix multiplication andcbe a scalar, then the following properties hold.

1.A(BC) = (AB)CAssociative property of multiplication

2.A(B+C) =AB+ACDistributive property

3. (A+B)C=AC+BCDistributive property

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

{ Commutativity:In general,AB6=BA!. Note that ifABis dened,BAmay not be dened. { Cancelation:IfAC=BC, you can NOT sayA=B. You can not simply cancel like in scalar multiplication. There are conditions onCwhich must be met in order to apply the cancelation principle. We will discuss these conditions in a later section. As an example, given A=2

41 2 3

0 5 4 32 13
5 ;B=2 446 3
5 4 4

1 0 13

5 ;C=2

40 0 0

0 0 0 42 33
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