1 Permutations and Combinations CS/APMA 202 Epp section 6.4 Aaron Bloomfield.
SECTION 6.4
description
Transcript of SECTION 6.4
SECTION 6.4SECTION 6.4
MATRIX MATRIX ALGEBRAALGEBRA
THE ALGEBRA OF MATRICES
THE ALGEBRA OF MATRICES
Addition:Addition:
39-
4-5
512
31-
83
1-4
PROPERTIES OF MATRICES
PROPERTIES OF MATRICES
Commutative:Commutative:
A + B = B + AA + B = B + A
Associative:Associative:
A + (B + C) = (A + B) + A + (B + C) = (A + B) + CC
ZERO FOR MATRICESZERO FOR MATRICES
00
000
ADDITIVE INVERSE FOR MATRICES
ADDITIVE INVERSE FOR MATRICES
dc
ba A If
d-c-
b-a- A- then
MORE PROPERTIES OF MATRICES
MORE PROPERTIES OF MATRICES
Additive IdentityAdditive Identity:: There is a There is a matrix 0 satisfyingmatrix 0 satisfying
0 + A = A + 0 = A0 + A = A + 0 = A
Additive InversesAdditive Inverses:: For each For each matrix A, there is a matrix -A matrix A, there is a matrix -A satisfyingsatisfying
A + (-A) = (-A) + A = 0A + (-A) = (-A) + A = 0
MORE PROPERTIES OF MATRICES
MORE PROPERTIES OF MATRICES
k(A + B) = kA + kBk(A + B) = kA + kB
(k + m)A = kA + mA(k + m)A = kA + mA
(km)A = k(mA) = m(kA)(km)A = k(mA) = m(kA)
5-3-
2-7 B and
25
63- A
3A 3A = =
615
189- -2B -2B
= =
106
414-
3A - 2B = 3A - 2B =
1621
2223-
MULTIPLICATION:MULTIPLICATION:
DC
BA
dc
ba
DbC
BaA
aA+baA+bCC
bDC
aBA
aB+baB+bDD
MULTIPLICATION:MULTIPLICATION:
DC
BA
dc
ba
DdC
BcA
aA+baA+bCC
cA+dCcA+dC
dDC
cBA
aB+baB+bDD
cB+dcB+dDD
43
25- B
41-
3-2 A
Find A Find A B B
43(3)-
2-5)(2
- 19- 19
3(4)-3
)2(25-
- 8- 8
43
25- B
41-
3-2 A
44(3)
2-5)(1-
- 19- 19
Find A Find A B B
1717
4(4)3
)2(1-5-
- 8- 8
1414
Find B Find B A A
42(-1)
3-)2(5-
- 12- 12
2(4)1-
5(-3)-2-
2323
41-
3-2 A
43
25- B
44(-1)
3-)2(3
- 12- 12
22
4(4)1-
3(-3)2-
2323
77
Find B Find B A A
41-
3-2 A
43
25- B
MULTIPLICATION:MULTIPLICATION:
AB AB = =
1417
8-19-
BA BA = =
72
2312-
AB AB BA BA
MultiplicatioMultiplication is not n is not
commutativcommutative.e.
MORE PROPERTIES OF MATRICES
MORE PROPERTIES OF MATRICES
Associative:Associative:
A A (B (B C) = (A C) = (A B) B) C C
Distributive:Distributive:
A A (B + C) = A (B + C) = A B + A B + A C C
(B + C) (B + C) A = B A = B A + C A + C A A
COMPATABILITY OF MATRICES
COMPATABILITY OF MATRICES
For Addition:For Addition: Same SizeSame Size
Counterexample:Counterexample:
241-
43-2 A
32-
25 B
COMPATABILITY OF MATRICES
COMPATABILITY OF MATRICES
For Multiplication:For Multiplication:
241-
43-2 A
32-
25 B
2-1-4
14-2
031
C
43-
1-2
2-3
D
COMPATABILITY OF MATRICES
COMPATABILITY OF MATRICES
For Multiplication:For Multiplication:
Two matrices are compatable Two matrices are compatable for multiplication when the for multiplication when the dimensions are:dimensions are: n x m and n x m and m x p m x p
The product matrix will have The product matrix will have dimension n x pdimension n x p
USING A CALCULATOR
USING A CALCULATOR
For Multiplication:For Multiplication:
241-
43-2 A
32-
25 B
2-1-4
14-2
031
C
43-
1-2
2-3
D
INVERSES OF MATRICES
INVERSES OF MATRICES
With matrices, when AB = With matrices, when AB = I I = = BA, we say A and B are BA, we say A and B are inverses.inverses.
Furthermore, when a matrix A Furthermore, when a matrix A has an inverse, we denote it as has an inverse, we denote it as AA -1 -1
Many matrices will fail to have Many matrices will fail to have an inverse.an inverse.
THEOREM: MULTIPLICATIVE
INVERSES
THEOREM: MULTIPLICATIVE
INVERSES The matrixThe matrix
dc
ba A
has a multiplicative inverse if has a multiplicative inverse if and only if D = ad - bc is and only if D = ad - bc is nonzero.nonzero.
dc
ba A
If D If D 0, 0, thenthen
Da
Dc-
Db-
Dd
A 1 -
9-5
5-2 A
Does A have an Does A have an inverse?inverse?D = 2(-9) - D = 2(-9) - 5(-5) 5(-5)
= -18 + = -18 + 25 25
= 7= 7
72
75-
75
79-
A 1 -
106-
5-3 A
Does A have an inverse?Does A have an inverse?
D = 3(10) - (-6)(-D = 3(10) - (-6)(-5) 5)
= 30 - 30 = 30 - 30
= 0= 0
INVERSES OF MATRICES
INVERSES OF MATRICES
This method of finding the This method of finding the inverse of a matrix works inverse of a matrix works nicely on 2 x 2 matrices.nicely on 2 x 2 matrices.
Finding an inverse of a 3 x 3 Finding an inverse of a 3 x 3 matrix is another matter.matrix is another matter.
We’ll use the calculator!We’ll use the calculator!
EXAMPLE:EXAMPLE:
772
672
662
A
11-0
011-
3-027
A 1 -
APPLICATIONS TO SYSTEMS OF EQUATIONS
APPLICATIONS TO SYSTEMS OF EQUATIONS
2x + 6y + 6z = 22x + 6y + 6z = 22x + 7y + 6z = - 32x + 7y + 6z = - 32x + 7y + 7z = - 52x + 7y + 7z = - 5
772
672
662
z
y
x
= =
5-
3-
2
APPLICATIONS TO SYSTEMS OF EQUATIONS
APPLICATIONS TO SYSTEMS OF EQUATIONS
772
672
662
z
y
x
= =
5-
3-
2AA- 1- 1 AA- 1- 1
z
y
x
= =
AA- 1- 1
5-
3-
2
SOLVE THE SYSTEM OF EQUATIONS
SOLVE THE SYSTEM OF EQUATIONS
2x + 6y + 6z = 22x + 6y + 6z = 22x + 7y + 6z = - 32x + 7y + 6z = - 32x + 7y + 7z = - 52x + 7y + 7z = - 5
z
y
x
= =
772
672
662
- 1- 1
5-
3-
2
SOLVE THE SYSTEM OF EQUATIONS
SOLVE THE SYSTEM OF EQUATIONS
z
y
x
= =
2-
5-
22
CONCLUSION OF SECTION 6.4CONCLUSION OF SECTION 6.4