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1 Operations with Matrice
2 Properties of Matrix Operations
3 The Inverse of a Matrix
4 Elementary Matrices
5 Applications of Matrix Operations
2.1
2.2
Matrix:
nm
nmmnmmm
n
n
n
ij M
aaaa
aaaa
aaaa
aaaa
aA
][
321
3333231
2232221
1131211
(i, j)-th entry (or element): ija
number of rows: m
number of columns: n
size: m×n
Square matrix: m = n
2.3
For [ ] and [ ] ,ij m n ij m nA a B b
Equal matrices: two matrices are equal if they have the same size (m × n) and entries corresponding to the same position are equal
if and only if for 1 , 1ij ijA B a b i m j n
Ex 1: Equality of matrices
dc
baBA
43
21
If , then 1, 2, 3, and 4A B a b c d
2.4
Matrix addition:
If [ ] , [ ] ,ij m n ij m nA a B b
then [ ] [ ] [ ] [ ]ij m n ij m n ij ij m n ij m nA B a b a b c C
Ex 2: Matrix addition
31
50
2110
3211
21
31
10
21
2
3
1
2
3
1
22
33
11
0
0
0
2.5
Matrix subtraction:
BABA )1(
Scalar multiplication:
If [ ] and is a constant scalar,ij m nA a c
Ex 3: Scalar multiplication and matrix subtraction
212
103
421
A
231
341
002
B
Find (a) 3A, (b) –B, (c) 3A – B
then [ ]ij m ncA ca
2.6
(a)
212
103
421
33A
(b)
231
341
002
1B
(c)
231
341
002
636
309
1263
3 BA
Sol:
636
309
1263
231323
130333
432313
231
341
002
407
6410
1261
2.7
Matrix multiplication:If [ ] and [ ] ,ij m n ij n pA a B b
then [ ] [ ] [ ] ,ij m n ij n p ij m pAB a b c C
njin
n
kjijikjikij babababac
1
2211where
size of C=AB
inijii
nnnjn
nj
nj
nnnn
inii
n
cccc
bbb
bbb
bbb
aaa
aaa
aaa
21
1
2221
1111
21
21
11211
※ The entry cij is obtained by calculating the sum of the entry-by-entry product between the ith row of A and the jth column of B
should be equal
2.8
3 2
1 3
4 2
5 0
A
2 2
3 2
4 1B
Ex 4: Find AB
Sol:
3 2
3 2
( 1)( 3) (3)( 4) ( 1)(2) (3)(1)
(4)( 3) ( 2)( 4) (4)(2) ( 2)(1)
(5)( 3) (0)( 4) (5)(2) (0)(1)
9 1
4 6
15 10
AB
Note: (1) BA is not multipliable
(2) Even BA is multipliable, AB≠BA
2.9
Matrix form of a system of linear equations in n variables:
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
2211
22222121
11212111
= = =
A x b
equationslinear m
single matrix equation
A x b1 nnm 1m
mnmnmm
n
n
b
b
b
x
x
x
aaa
aaa
aaa
2
1
2
1
21
22221
11211
2.10
Diagonal matrix : a square matrix in which nonzero elements are found only in the principal diagonal
),,,( 21 nddddiagA nn
n
M
d
d
d
00
00
00
2
1
※ It is the usual notation for a diagonal matrix.
Keywords in Section 2.1:equality of matrices:
matrix addition:
scalar multiplication:
matrix multiplication:
row vector:
column vector:
diagonal matrix:
2.11
2.12
2.2 Properties of Matrix Operations
Three basic matrix operators, introduced in Sec. 2.1:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
Zero matrix :
0 0 0
0 0 0
0 0 0
m n
m n
0
Identity matrix of order n :
1 0 0
0 1 0
0 0 1
n
n n
I
2.13
then (1) A+B = B+A
(2) A+(B+C) = (A+B)+C
(3) ( cd ) A = c ( dA )
(4) 1A = A
(5) c( A+B ) = cA + cB
(6) ( c+d ) A = cA + dA
If , , , and , are scalars,m nA B C M c d
Properties of matrix addition and scalar multiplication:
(Commutative property of addition)
(Associative property of addition)
(Associative property of scalar multiplication)
(Multiplicative identity property, and 1 is the multiplicative identity for all matrices)
(Distributive property of scalar multiplication over matrix addition)
(Distributive property of scalar multiplication over real-number addition)
Notes:
All above properties are very similar to the counterpart properties for real numbers
2.14
If , and is a scalar,m nA M c
then (1) m nA A 0
(2) ( ) m n A A 0
(3) 0 or m n m n cA c A 0 0
Notes:
All above properties are very similar to the counterpart properties for the real number 0
Properties of zero matrices:
※ So, 0m×n is also called the additive identity for the set of all m×n matrices
※ Thus , –A is called the additive inverse of A
2.15
(1) A (BC) = (AB ) C
(2) A (B+C) = AB + AC
(3) (A+B)C = AC + BC
(4) c (AB) = (cA) B = A (cB)
Properties of the identity matrix:
If , then (1)
(2) m n n
m
A M AI A
I A A
Properties of matrix multiplication:(Associative property of matrix multiplication)
(Distributive property of LHS matrix multiplication over matrix addition)(Distributive property of RHS matrix multiplication over matrix addition)
※ For real numbers, the properties (2) and (3) are the same since the order for the multiplication of real numbers is irrelevant.
※ For real numbers, in addition to satisfying above properties, there is a commutative property of real-number multiplication, i.e., cd = dc.
※ For real numbers, the role of 1 is similar to the identity matrix. However, 1 is unique for real numbers and there could be many identity matrices with different sizes
2.16
1 01 2 1 0 2
, , and 3 1 .2 1 3 2 1
2 4
A B C
Ex 3: Matrix Multiplication is Associative
Calculate (AB)C and A(BC) for
Sol:1 0
1 2 1 0 2( ) 3 1
2 1 3 2 12 4
1 05 4 0 17 4
3 11 2 3 13 14
2 4
AB C
2.17
1 01 2 1 0 2
( ) 3 12 1 3 2 1
2 4
1 2 3 8 17 4
2 1 7 2 13 14
A BC
2.18
Properties for Ak:
Definition of Ak : repeated multiplication of a square matrix:
1 2
matrices
, , , k
k
A A A AA A AA A
(1) AjAk = Aj+k
(2) (Aj)k = Ajk
where j and k are nonegative integers and A0 is assumed to be I
1 1
2 2
0 0 0 0
0 0 0 0
0 0 0 0
k
kk
kn n
d d
d dD D
d d
For diagonal matrices:
2.19
11 12 1
21 22 2
1 2
If ,
n
nm n
m m mn
a a a
a a aA M
a a a
11 21 1
12 22 2
1 2
then
m
mTn m
n n mn
a a a
a a aA M
a a a
Transpose of a matrix :
※ The transpose operation is to move the entry aij (original at the position (i, j)) to the position (j, i)
※ Note that after performing the transpose operation, AT is with the size n×m
2.20
8
2A (b)
987
654
321
A (c)
11
42
10
A
Sol: (a)
8
2A 82 TA
(b)
987
654
321
A
963
852
741TA
(c)
11
42
10
A
141
120TA
(a)
Ex 8: Find the transpose of the following matrix
2.21
AA TT )( )1(TTT BABA )( )2(
)()( )3( TT AccA
)( )4( TTT ABAB
Properties of transposes:
※ Properties (2) and (4) can be generalized to the sum or product of multiple matrices. For example, (A+B+C)T = AT+BT+CT and (ABC)T = CTBTAT
※ Since a real number also can be viewed as a 1 × 1 matrix, the transpose of a real number is itself, that is, for , aT = a. In other words, transpose operation has actually no function on real numbers
a R
2.22
2 1 2
1 0 3
0 2 1
A
3 1
2 1
3 0
B
Sol:
Ex 9: Show that (AB)T and BTAT are equal
2 1 2 3 1 2 12 6 1
( ) 1 0 3 2 1 6 11 1 2
0 2 1 3 0 1 2
T T
TAB
2 1 03 2 3 2 6 1
1 0 21 1 0 1 1 2
2 3 1
T TB A
2.23
A square matrix A is symmetric if A = AT
Ex:
6
54
321
If
cb
aA is symmetric, find a, b, c?
A square matrix A is skew-symmetric if AT = –A
Skew-symmetric matrix :
Sol:
6
54
321
cb
aA
653
42
1
c
ba
AT
5 ,3 ,2 cba
TAA
Symmetric matrix :
2.24
0
30
210
If
cb
aA is a skew-symmetric, find a, b, c?
Note: TAA must be symmetric
Pf:
symmetric is
)()(T
TTTTTT
AA
AAAAAA
Sol:
030210
cbaA
032
01
0
c
ba
AT
TAA 3 ,2 ,1 cba
Ex:
※ The matrix A could be with any size, i.e., it is not necessary for A to be a square matrix.
※ In fact, AAT must be a square matrix.
2.25
ab = ba (Commutative property of real-number multiplication)
If , then is defined, but is undefinedm p AB BA(1)
mmmm MBAMABnpm , (3) then , If
If , , then , m m n nm p m n AB M BA M (2) (Sizes are not the same)
(Sizes are the same, but resultant matrices are not equal)
Real number:
Matrix:
BAAB pnnm
Three situations for BA:
n p m n
Before finishing this section, two properties will be discussed, which is held for real numbers, but not for matrices: the first is the commutative property of matrix multiplication and the second is the cancellation law
2.26
12
31A
20
12B
Sol:
44
52
20
12
12
31AB
BAAB
24
70
12
31
20
12BA
Ex 4 :
Sow that AB and BA are not equal for the matrices.
and
(noncommutativity of matrix multiplication)
2.27
Notes:
(1) A+B = B+A (the commutative law of matrix addition)
(2) (the matrix multiplication is not with the
commutative law) (so the order of matrix multiplication is very
important)
BAAB
※ This property is different from the property for the multiplication operations of real numbers, for which the order of multiplication is with no difference
2.28
(Cancellation law is not necessary to be valid)
0 , cbcac
b a (Cancellation law for real numbers)
Matrix: and ( is not a zero matrix)AC BC C C 0
(1) If C is invertible, then A = B
(2) If C is not invertible, then
Real number:
BA
※ Here I skip to introduce the definition of “invertible” because we will study it soon in the next section
2.29
21
21 ,
32
42 ,
10
31CBA
Sol:
21
42
21
21
10
31AC
So, although , BCAC BA
21
42
21
21
32
42BC
Ex 5: (An example in which cancellation is not valid)
Show that AC=BC
Keywords in Section 2.2:zero matrix:
identity matrix:
commutative property:
associative property: distributive property: cancellation law:
transpose matrix:
symmetric matrix:
skew-symmetric matrix: 2.30
2.31
2.3 The Inverse of a Matrix
nnMA
if there exists a matrix such that ,n n nB M AB BA I
Note:
A square matrix that does not have an inverse is called noninvertible (or singular )
Consider ,
then (1) A is invertible (or nonsingular )
(2) B is the inverse of A
Inverse matrix :
※ The definition of the inverse of a matrix is similar to that of the inverse of a scalar, i.e., c · (1/c) = 1
※ Since there is no inverse (or said multiplicative inverse for the real number 0, you can “imagine” that noninvertible matrices act a similar role to the real number 0 is some sense
2.32
If B and C are both inverses of the matrix A, then B = C.
Pf:
CB
CIB
CBCA
CIABC
IAB
)(
)(
Consequently, the inverse of a matrix is unique.
Notes:(1) The inverse of A is denoted by 1A
IAAAA 11 )2(
Theorem 2.7: The inverse of a matrix is unique
(associative property of matrix multiplication and the property for the identity matrix)
2.33
Gauss-Jordan Elimination 1| |A I I A
Ex 2: Find the inverse of the matrix A
31
41A
Sol:AX I
10
01
31
41
2221
1211
xx
xx
10
01
33
44
22122111
22122111
xxxx
xxxx
Find the inverse of a matrix by the Gauss-Jordan Elimination:
2.34
(1) ( 4)1,2 2,1, 1 4 1 1 0 3
(1)1 3 0 0 1 1
A A
(1) ( 4)1,2 2,1, 1 4 0 1 0 4
(2)1 3 1 0 1 1
A A
1 ,3 2111 xx
1 ,4 2212 xx
11
431AX
Thus
(2) 1304
(1) 0314
2212
2212
2111
2111
xxxx
xxxx
by equating corresponding entries
This two systems of linear equations have the same coefficient matrix, which is exactly the matrix A.
Perform the Gauss-Jordan elimination on the matrix A with the same row operations
2.35
(1) ( 4)1,2 2,1
Gauss Jordan Elimination
,
1
1 4 1 0 1 0 3 4
1 3 0 1 0 1 1 1
A A
A I I A
※ If A cannot be row reduced to I, then A is singular
Note:
Rather than solve the two systems separately, you can solve them simultaneously by adjoining (appending) the identity matrix to the right of the coefficient matrix
11
21
solution for x
x
12
22
solution for x
x
2.36
326101011
A
( 1)1,2
1 1 0 1 0 0
0 1 1 1 1 0
6 2 3 0 0 1
A
Sol:
100010001
326101011
IA
(6)1,3
1 1 0 1 0 0
0 1 1 1 1 0
0 4 3 6 0 1
A
( 1)3
1 1 0 1 0 0
0 1 1 1 1 0
0 0 1 2 4 1
M
( 4)2,3
1 1 0 1 0 0
0 1 1 1 1 0
0 0 1 2 4 1
A
Ex 3: Find the inverse of the following matrix
2.37
(1)3,2
1 1 0 1 0 0
0 1 0 3 3 1
0 0 1 2 4 1
A
(1)2,1
1 0 0 2 3 1
0 1 0 3 3 1
0 0 1 1 4 1
A
So the matrix A is invertible, and its inverse is
142
133
1321A
] [ 1 AI
Check it by yourselves:
IAAAA 11
2.38
If A is an invertible matrix, k is a positive integer, and c is a scalar,
then AAA 111 )( and invertible is (1)
1 1(2) is invertible and ( ) ( )k k k kA A A A
1 11(3) c is invertible if 0 and ( )A c cA A
c
Theorem 2.8: Properties of inverse matrices
TTT AAA )()( and invertible is (4) 11
1
1 1
Ex.
2 3 2 4 0.1 0.3
4 1 3 1 0.4 0.2
0.1 0.4( ) ( )
0.3 0.2
T
T T
A A A
A A
← “T” is not the number of power. It denotes the transpose operation
2.39
Theorem 2.9: The inverse of a product If A and B are invertible matrices of order n, then AB is invertible and
111)( ABAB
1 1Thus, if is invertible, then its inverse is AB B A
Pf: 1 1 1 1 1 1 1( )( ) ( ) ( ) ( )AB B A A BB A A I A AI A AA I
Note:
(1) It can be generalized to the product of multiple matrices
(2) It is similar to the results of the transpose of the products of multiple matrices (see Slide 2.23)
11
12
13
11321
AAAAAAAA nn
1 2 3 3 2 1
T T T T Tn nA A A A A A A A
(associative property of matrix multiplication)
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