Inverse and Identity Matrices
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Transcript of Inverse and Identity Matrices
Inverses and Identities
5x = 351
51
531 x53x
inversestivemultiplicaareand 551
identitytivemultiplicatheis1
Jeff Bivin -- LZHS
1001
Now with Matrices
4321
4321
This is theIdentity Matrix
for 2 x 2 Matrices
Jeff Bivin -- LZHS
Let’s look at another example
5387
1001
53
87
Jeff Bivin -- LZHS
New Question
4321
1001
What do we multiply a matrix
by to get the Identity?
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
dcba
dcba1
acbd
0dcba
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
4321
43211
1324
1324
64121
21
23
12
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
2134
2134
1
41
32
41
32381111
114
111
113
112
Jeff Bivin -- LZHS
:' usesLet
3125
4321X
BAX BAX 1A
BAXI 1
1A
BAX 1This is our Formula!
3125
1324
43211X21
9141418
21X
29779
XJeff Bivin -- LZHS
:' usesLet
3427
5431
X
BXA BXA 1A
1 ABIX
1A
1 ABXThis is our Formula!
14
353427
5431
1X
15322343
71X
715
732
723
743
71
1435
3427
71X
Jeff Bivin -- LZHS
:' usesLet
6314
4132
3251X
CBAX
)( BCAX 1A BCAXI 1
1A
BCAX 1
This is our Formula!
41
326314
1253
3251
1X71
2442
1253
71X
68226
71X
BCAX
76
78
72
726
Jeff Bivin -- LZHS
Are the two Matrices Inverses?
8532
2538
1001
Jeff Bivin -- LZHS
The product of inverse matrices
is the identity matrix.
Identity, therefore, INVERSEMatrices
Are the two Matrices Inverses?
4123
3124
100010
Jeff Bivin -- LZHS
The product of inverse matrices
is the identity matrix.
Not the Identity,
therefore, NOT
INVERSEMatrices
4623
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
dcba
dcba1
acbd
0dcba
Jeff Bivin -- LZHS
4623
Find the determinant!
4623
Jeff Bivin -- LZHS
01212
Therefore, NO inverse!
Does the Matrix have an Inverse?
14427
Find the determinant!
14427
Jeff Bivin -- LZHS
90898
Therefore, an inverse exists!
Does the Matrix have an Inverse?
987654321
Find the determinant!
Jeff Bivin -- LZHS
Therefore, NO inverse!
Does the Matrix have an Inverse?
1 2 3 1 24 5 6 4 57 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
243142231
Find the determinant!
Jeff Bivin -- LZHS
Does the Matrix have an Inverse?
1 3 2 1 32 4 1 2 43 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
75423
yx
BAU BAU 1A
BAUI 1
1A
BAU 1This is our Formula!
117
3425
54231U 231
557
231U
2352357
Jeff Bivin -- LZHS
3x + 2y = 74x - 5y = 11
Solve the system using inverse matrices
235
2357 , solution
:' usesLet
19
2342
yx
BAU BAU 1A
BAUI 1
1A
BAU 1This is our Formula!
1
92342
2342
1U 81
2514
81U
82547
Jeff Bivin -- LZHS
2x - 4y = 93x - 2y = 1
Solve the system using inverse matrices
825
47 , solution
