matrices jeffrey bivin lake zurich high school [email protected] last updated: october 12, 2005
TRANSCRIPT
Inverses and Identities
5x = 351
51
531 x53x
inversestivemultiplicaareand 551
identitytivemultiplicatheis1
Jeff Bivin -- LZHS
10
01
Now with Matrices
43
21
43
21
This is theIdentity Matrix
for 2 x 2 Matrices
Jeff Bivin -- LZHS
Let’s look at another example
53
87
10
01
53
87
Jeff Bivin -- LZHS
New Question
43
21
10
01
What do we multiply a matrix
by to get the Identity?
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
dc
ba
dc
ba1
ac
bd
0dc
ba
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
43
21
43
211
13
24
13
2464
1
21
21
23
12
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
21
34
21
341
41
32
41
3238
1
111
114
111
113
112
Jeff Bivin -- LZHS
:' usesLet
31
25
43
21X
BAX BAX 1A
BAXI 1
1A
BAX 1This is our Formula!
31
25
13
24
43
211X2
1
914
141821X
297
79X
Jeff Bivin -- LZHS
:' usesLet
34
27
54
31X
BXA BXA 1A
1 ABIX
1A
1 ABX
This is our Formula!
14
35
34
27
54
311X
1532
2343
7
1X
715
732
723
743
7
1
14
35
34
27
7
1X
Jeff Bivin -- LZHS
:' usesLet
63
14
41
32
32
51X
CBAX
)( BCAX 1A
BCAXI 1
1A
BCAX 1
This is our Formula!
41
32
63
14
12
53
32
511X7
1
24
42
12
5371X
68
22671X
BCAX
76
78
72
726
Jeff Bivin -- LZHS
Are the two Matrices Inverses?
85
32
25
38
10
01
Jeff Bivin -- LZHS
The product of inverse matrices
is the identity matrix.
Identity, therefore, INVERSEMatrices
Are the two Matrices Inverses?
41
23
31
24
100
010
Jeff Bivin -- LZHS
The product of inverse matrices
is the identity matrix.
Not the Identity,
therefore, NOT
INVERSEMatrices
46
23
Jeff Bivin -- LZHS
Does the Matrix have an Inverse?
Let’s review the definition of the Inverse of a
2x2 Matrix
The Inverse of a 2x2 Matrix
dc
ba
dc
ba1
ac
bd
0dc
ba
Jeff Bivin -- LZHS
46
23Find the determinant!
46
23
Jeff Bivin -- LZHS
01212
Therefore, NO inverse!
Does the Matrix have an Inverse?
144
27Find the determinant!
144
27
Jeff Bivin -- LZHS
90898
Therefore, an inverse exists!
Does the Matrix have an Inverse?
987
654
321
Find the determinant!
Jeff Bivin -- LZHS
Therefore, NO inverse!
Does the Matrix have an Inverse?
1 2 3 1 2
4 5 6 4 5
7 8 9 7 8
1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2
45 + 84 + 96 - 105 - 48 - 72
243
142
231
Find the determinant!
Jeff Bivin -- LZHS
Does the Matrix have an Inverse?
1 3 2 1 3
2 4 1 2 4
3 4 2 3 4
1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3
8 + 9 + 16 - 24 - 4 - 12
Therefore, an inverse exists!
:' usesLet
11
7
54
23
y
x
BAU BAU 1A
BAUI 1
1A
BAU 1This is our Formula!
11
7
34
25
54
231U 23
1
5
57231U
235
2357
Jeff Bivin -- LZHS
3x + 2y = 74x - 5y = 11
Solve the system using inverse matrices
235
2357 , solution
:' usesLet
1
9
23
42
y
x
BAU BAU 1A
BAUI 1
1A
BAU 1This is our Formula!
1
9
23
42
23
421U 8
1
25
1481U
825
47
Jeff Bivin -- LZHS
2x - 4y = 93x - 2y = 1
Solve the system using inverse matrices
825
47 , solution