Sec 3.5 Inverses of Matrices
1
A* A  I
Where A is nxn
Finding the inverse of A:
A
I
Seq or row operations
I
A
1

Finding the inverse of A:
A
* * * 1 0 0
* * * 0 1 0


* * * 0 0 1
1 * * * * *
* * * * * *


* * * * * *
1 * * * * *
0 * * * * *


0 * * * * *
1 * * * * *
0 1 * * * *


0 0 1 * * *
1 * * * * *
0 1 * * * *


0 0 * * * *
1 * * * * *
0 1 * * * *


0 * * * * *
1 * 0 * * *
0 1 0 * * *


0 0 1 * * *
1 0 0 * * *
0 1 0 * * *


0 0 1 * * *
Example
Find inverse
A
1
1 1 5 
A  1 4 13
3 2 12
Properties
1) A  B  B  A
2) A  ( B  C )  ( A  B)  C
3) A( BC )  ( AB)C
4) A( B  C )  AB  AC
5) ( A  B)C  AC  BC
Fact1: AB in terms of columns of B
B  [b1 , b2 ,, bn ]
AB  [ Ab1 , Ab2 ,, Abn ]
Fact1: Ax in terms of columns of A
x  [ x1 , x2 ,, xn ]
T
Ax  x1a1  x2 a2    x2 an
Basic unit vector:
1
0 
e1   ,

 
0 
0 
1
e2   , 

 
0 
Ae j  ????
AI  ????
0 

 
e j  1 
 

0
J-th location
Def: A is invertable if
There exists a matrix B such that
AB  BA  I
TH1: the invers is unique
TH2: the invers of 2x2 matrix
1  d  b
A
det  c a 
det  ad  bc
a b 
A

c
d


Example
Find inverse
 4 6
A

5
9


TH3: Algebra of inverse
If A and B are invertible, then
1
A1 is invertable and ( A1 )-1  A
2
n  0, An is invertible and ( An ) 1  ( A1 ) n
3
4
( AB) 1  B 1 A1
( ABCD) 1  D 1C 1B 1 A1
TH4: solution of Ax = b
A is nxn invertible matrix and b is a n - vector
Ax  b
has a uique sol x  A1b
Example
4x  6 y  6
5 x  9 y  18
Solve
Def: E is elementary matrix if
1) Square matrix nxn
2) Obtained from I by a single row operation
1 0
I 

0 1
1 0 0
I  0 1 0
0 0 1
R1  R2
2 R1  R3
0 1
E1  

1
0


1 0 0 
E2  0 1 0


2 0 1
Which of these matrices are elementary matrices
 5 1
E1  

1
0


1
0
E5  
0

0
0 0 0
1 0 0

0 1 1

0 0 1
1  2 0
E2   0 1 0 


0 0 1
1
1
E5  
0

0
0 0 0
1 0 0

0 1 0

0 1 1
1  2 0
E3  0 0 1


0 1 0
0
0
E4  
0

1
0 0 1
1 0 0

0 1 0

0 0 0
REMARK:
Let E corresponds to a certain elem row operation.
It turns out that if we perform this same operation on matrix A , we
get the product matrix EA
2 R1  R3
1 0 0
I  0 1 0
0 0 1
 1 1 2
A   3 1 4
 2 0 1
2 R1  R3
1 1 2
A2  0 2 5


3 1 4
 ??    ??
R2  R3
1 0 0
E2  0 0 1


0 1 0
1 0 0 
E1  0 1 0


2 0 1
1 1 2
A1  3 1 4
0 2 5
1 1 2
A1  3 1 4
0 2 5
A1  E1 A
NOTE:
1 0 0
I  0 1 0
0 0 1
E11
Every elementary matrix is invertible
R1  R3
 ??
1 0 0
I  0 1 0
0 0 1
1
E3
0 0 1 
E1  0 1 0


1 0 0
 ??
0 0 1 
E11  0 1 0


1 0 0
1 0 0
I  0 1 0
0 0 1
E21  ??
TypeI
1 0 0 
A1  0 1 0


2 0 1
2 R1  R3
1 0 0 
E3  0 1 0


2 0 1
1
E21  0

0
I  Eˆ 3 E3
0
1
5
0
0
0

1
5R2
 2 R1  R3
 1 0 0
Eˆ 3   0 1 0


 2 0 1
 1 0 0
E31   0 1 0


 2 0 1
1 0 0
E2  0 5 0


0 0 1
TypeII
1 0 0
I  0 1 0
0 0 1
TypeIII
Any observation??
Sec 3.5 Inverses of Matrices
TH6:
A
A
identity matrix I
is invertible
A
I
Row operation 1
A1
A1  E1 A
Row operation 2
*
A2
*
Row operation 3

Row operation k
Ak 1  Ek 1 Ak 2
A2  E2 A1
I  Ek Ek 1  E2 E1 A
1
A  Ek Ek 1  E2 E1
A  Ek Ek 1  E2 E1 
1
A
is row equivalent to
1 1
1 1
E1 E2  Ek 1Ek
I
A1

I  Ek Ak 1
Note : the sequence of elementary row
operations that tran sform A into I also
transforms I into A -1
A is
invertable
A is a product of
elementary matrices
Solving linear system
Example
Solve
4x  3y  2z  2
5 x  6 y  3z  1
3x  5 y  2 z  4
 4 3 2  x   2 
5 6 3  y   1

   
3 5 2  z  4
 x
2  14 
 3 4 3 
 y   A1 1   8 


1

A   1 2 2 
 
  

 z 
4  39
 7 11  9
Example
Solve
4x  3y  2z  0
5 x  6 y  3z  0
3x  5 y  2 z  0
 4 3 2   x  0 
 5 6 3   y   0 

   
3 5 2  z  0
What is the
solution
Quiz2: SAT in Class (3.3+3.4)
1) Given a matrix A find the reduced row echelon form
 2 1 2 0
A  1 1 0 0 


3  1 1 0
2) Use the method of Gauss-Jordan elimination to solve
the following system (find the solution in vector form (i.e) as
a linear combination of vectors)
1 9 0 0 0
0 0 1 0 0 


0 0 0 1 0
Quiz3: Sund online (3.3+3.4)
Matrix Equation
In certain applications, one need to solve a system Ax = b of n
equations in n unknowns several times but with different vectors b1, b2,..
Ax1  b1
Example
 4 3 2
A  5 6 3
3 5 2
Ax2  b2  Axk  bk
Solve
AX  B
 3  1 2 6
B  7 4 1 5
5 2 4 1
 3 4 3 
A1   1  2 2 
 7 11  9
1
XA B
Ax1
Ax1
Ax2 
Axk   b1
x2  xk   b1
AX  B
b2  bk 
b2  bk 
Matrix Equation
Definition:
A is nonsingular matrix if
the trivial solution
Example
Show that A is
nonsingular
 1 1 0
A   0 1 0
 1 0 1
the system
Ax  0
has only
x0
RECALL: Definitions
invertible
nonsingular
Row equivalent
Theorem7:(p193)
A
row equivalent
I
Every n-vector b
Ax = b
has unique sol
A
Invertible
A
is a product of
elementary
matrices
Every n-vector b
Ax = b
is consistent
The system
Ax = 0
has only the
trivial sol
A
nonsingular
TH7:
A is an nxn matrix. The following is equivalent
(a) A is invertible
(b) A is row equivalent to the nxn identity matrix I
(c) Ax = 0 has the trivial solution
(d) For every n-vector b, the system A x = b has a unique solution
(e) For every n-vector b, the system A x = b is consistent
Problems (page194)
34) Show that a diagonal matrix is inverible if and only if
each diagonal element is nonzero. In this case , state
concisely how the invers matrix is obtained.
35) Let A be an nxn matrix with either a row or a column
consisting only of zeros. Show that A is not invertible.
41) Show that the i-th row of the product AB
where Ai is the i-th row of the matrix A.
is Ai B,
?
?
?
? ? ?
True & False
A
row equivalent
I
Every n-vector b
Ax = b
has unique sol
A
not Invertible
A
is a product of
elementary
matrices
Every n-vector b
Ax = b
is consistent
The system
Ax = 0
has only the
trivial sol
A
nonsingular