Post on 02-Jun-2018
8/10/2019 LinAlg-Chapter2 part1 s1.ppt
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Chapter 2 part12.1 2.2 2.3
Matrices
Linear Algebra
S 1
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Ch2_2
2.1 Ad dit ion, Scalar Mu lt ip l icat ion, and Mult ip l icat ion o f Matr ices
Definition
Amatrix is a rectangular array of numbers.
The numbers in the array are called the elementsof the matrix.
letter.capitalDenoted by: A,B,
11 12 1
21 22 2
1 2
m
m
n n nm
a a a
a a aA
a a a
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Ch2 3
2.1 Ad dit ion, Scalar Mu lt ip l icat ion, and Mult ip l icat ion o f Matr ices
aij: the element of matrix Ain row.. and column
we say it is in the ..
The sizeof a matrix = number of ... number of ... = .
If n=m the matrix is said to be a .. matrix with size = ...or .
A matrix that has one row is called a matrix. A matrix that has one columnis called a ..matrix.
For a square nnmatrixA, the main diagonalis: .
We can denote the matrix by ..
Note:
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Ch2_4
Definition
Two matrices are equalif:
1) ..
2) ..
Example 1
2 5
3 0
1) ................
2) ............... .3) .............. .
A
A size is
A is a matrixis the main diagonal
3 1 7 11 2 2 4
0 0 3 1
' .........
B
it s size is
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Ch2_5
Addition of Matrices
Definition
IfAandBbe matrices of the .. then the
sumA+B=C will be of the .. size and
If
LetAbe a matrix and k be a scalar. The scalar multipleofAby
k , denoted will be the same size asA.
The matrix (-1)A= -A called the of A.LetAandBof the same size then:A-B= A+(-B)=Cand:
..................A size B size A B
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Ch2_6
Example 2
.7245
and,813652
,320741
Let
CBA
DetermineA+B, 3A ,A+ C , A-B
Solution
3
(3
(1)
(2)
)
(4)
A
A
A B
C
A B
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Definition
A. matrix all of its elements are zero. If the zero
matrixis of a square size nn it will be denoted by .
0ij0
n
Theorem2.2:
Let A,B,C bematrices, be scalars.
Assume that the size of the matrices are such that the operations
can be performed, let 0 be the zero matrix.
Propert ies of matr ix addit ion and s calar mu lt ip l icat ion :
1 2,k k
1) ............
2) ...............
3) 0 0 ........
A B
A B C
A A
1
1 2
1 2
4) .............
5) .............
6) ................
k A B
k k C
k k A
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2 3 5A B C
Example 3
Compute the linear combination: for:
1 3 3 7 0 2, ,
4 5 2 1 3 1A B C
Solut ion
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Mult ip l icat ion o f Matr ices
Definition
1) If the number of .. in A= the number of .. inB.
The productABthen exists.
LetA: ..matrix,B: .matrix,
The product matrix C=AB is a .matrix.
2) If the number of .in A the number of ..inB
then The product AB ..
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11 12 1 11 12 1
21 22 2 21 22 2
1 2 1 2
11 12 1 11 12 1
21 22 2 21 22 2
1 2 1 2
,
m k
m k
n n nm m m mk
m k
m k
n n nm m m mk
a a a b b b
a a a b b bif A B
a a a b b b
a a a b b b
a a a b b bA B
a a a b b b
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..............................................ij
If AB C
then c
Note:
Examp le 4
Let C=AB, Determine c23.
105237
and4312
BA
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Example 5
3 1 2 7 2
Let , Determine , .4 1 5 6 3A B AB BA
Solut ion
Note.In general,
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Ch2_13
Definition
1) A ..matrixis a matrix in which all the elements are zeros.
2) A ..matrixis a square matrix in which all the
elements ...
3) An ..matrixis a diagonal matrix in which every
element in the main diagonal is .
matrixzero
000
000000
mn0
11
22
0 0
0 0
0 0
................. matrix A
nn
a
aA
a
1 0 0
0 1 0
0 0 1
............... matrix
nI
Special Matrices
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Ch2_14
Theorem 2.1
LetAbe mnmatrix and Omnbe the zero mnmatrix. LetBbe
an n
nsquare matrix. OnandInbe the zero and identity n
nmatrices. Then:
1) A+ Omn= Omn+A= .
2) BOn= OnB= 3) BIn=InB=
Examp le 6
.4312
and854312
Let
BA
23 ...............A O
2 .................BO
2 ...........BI
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Ch2_15
LetA,B, andCbe matrices and kbe a scalar. Assume that the size of
the matrices are such that the operations can be performed.
Properties of Matr ix Mul tipli cation
1.A(BC) = . Associative property of multiplication
2.A(B+ C) = Distributive property of multiplication
3. (A+B)C= Distributive property of multiplication
4.AIn=InA= (whereInis the identity matrix)
5. k(AB) = = Note:ABBA in general. Multiplication of matrices is not
commutative.
Theorem 2.2 -2
2.2 A lgebraic Propert ies of Matr ix Operat ions
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Ch2_16
Example 7
.
0
14
and,
201
310,
13
21Let
CBA ComputeABC.
Solut ion
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Ch2_17
In algebra we know that the following cancellation laws apply.
If ab= acand a 0 then ..Ifpq= 0 then ..or .
However the corresponding results are not true for matrices.
AB=AC.thatB= C.
PQ= Othat P= Oor Q= O.
Note:
Examp le 81 2 1 2 3 8
(1) Consider the matrices , , and .2 4 2 1 3 2
Observe that .................................., but ................
A B C
1 0 0 0(2) Consider the matrices , and .
0 0 0 1
Observe that ........................, but ........................
P Q
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Ch2_18
Powers of Matrices
Theorem 2.3
IfAis an nn square matrix andrandsare nonnegative
integers, then1. ArAs= .
2. (Ar)s= .
3. A0= (by definition)
Defini t ion
IfAis a squarematrix and kis a positive integer, then
...........................kA
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Ch2_19
Example 9
.compute,
01
21If 4AA
Solution
Example 10 Simplify the following matrix expression.
ABBABABBAA 57)2(3)2( 22
Solution
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Ch2_20
Idempo tent and Nilpotent Matr ices
Defini t ion
A square matrixAis said to be:
.if ..
.if there is a positive integerps.t .
The least integerpsuch thatAp=0 is called the
. of the matrix.
3 6
(1) 1 2A
Example 11
3 9(2)
1 3B
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Ch2_21
2.3 Symmetr ic Matr ices
Definition
The..of a matrixA, denoted , is the matrixwhose .. are the . of the given matrixA.
Example 12
.431and,654721
,0872
CBA
i.e., : : ..............tA m n A
Determine thetransposeof the following matrices:
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Ch2_22
Theorem :Propert ies of Transpose
LetAandBbe matrices and kbe a scalar. Assume that the sizes
of the matrices are such that the operations can be performed.
1. (A+B)t= ... Transpose of a sum
2. (kA)t= ... Transpose of a scalar multiple
3. (AB)t= ... Transpose of a product
4. (At)t= ...
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Ch2_23
Symmetr ic Matr ix
1 0 2 40 ....... 4
2 5 0 7 3 9
, 1 7 ....... ,5 4 2 3 2 3...... 8 3
4 9 3 6
match
match
Definition
Let A be a square matrix:
1) If ...then A called...matrix.
2) If ...then A called...matrix.
Example 13 symmetric matrices
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Ch2_24
Example 14
Proof
Let AandBbe symmetric matrices of the same size.
C = aA+bB, a,b are scalars. Provethat C is symmetric.
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Ch2_25
Example 15
Proof
Let AandBbe symmetric matrices of the same size.
Prove that the productABis symmetric if and only ifAB=BA.
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Example 16
Proof
Let Abe a symmetric matrix. Prove thatA2is symmetric.
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Definition
Let A be a square matrix. The of A denoted by ..isthe .of A.
LetAandBbe matrices and kbe a scalar. Assume that the sizesof the matrices are such that the operations can be performed.
1. tr(A + B) = ..
2. tr(kA) = .
3. tr(AB) =
4. tr(At) = ..
Theorem :Propert ies of Trace .