Download Matrix-Matrix Multiplication

Matrix-Matrix Multiplication
An Example.
3  2 1 

0  
 4 0  2 1 1 1
 1 2 0 3 1 0  2  





2
0 6
4 
3 = 12
0 
1 =
–2 
1 
0
1 = –2
0 =
0
10



Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10
 4 0  2 1 1 1
 1 2 0 3 1 0  2  





2
0 6
–1 
3 = –3
2 
1 =
2
0 
1 =
0
3 
0 =
0
–1



Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10
 4 0  2 1 1 1
 1 2 0 3 1 0  2   1





2
0 6
4 
–2 = –8
0 
1=
0
–2 
0=
0
1 
6=
6
–2



Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10  2
 4 0  2 1 1 1
 1 2 0 3 1 0  2   1





2
0 6
–1 
–2 =
2
2 
1=
2
0 
0=
0
3 
6 = 18
22



Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10  2
 4 0  2 1 1 1
 1 2 0 3 1 0  2   1 22





2
0 6
4 
1=
4
0 
0=
0
–2 
–2 =
4
1 
2=
2
10



Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10  2 10
 4 0  2 1 1 1
 1 2 0 3 1 0  2   1 22







2
0 6
–1 
1 = –1
2 
0=
0
0 
–2 =
0
3 
2=
6
5
Matrix-Matrix Multiplication
An Example.
3  2 1 

0  10  2 10
 4 0  2 1 1 1
 1 2 0 3 1 0  2   1 22 5 






2
0 6
Matrix-Matrix Multiplication
Another Example.
 2 3  2  x 
 4 2 3   y  

 
 5 7 6   z 
 2x + 3y – 2z


–4x + 2y + 3z
 5x + 7y + 6z
This gives us a simple and elegant way to write a system of linear equations.
 2 3  2
A   4 2 3 
 5 7 6 
 x
X   y 
 z 
  2
C   1 
 28 
2x + 3y – 2z = –2
Then the system
–4x + 2y + 3z = 1
5x + 7y + 6z = 28
can be written as AX  C.
Properties
ab  ba
AB  BA
Real-number multiplication is commutative.
Is matrix multiplication commutative?
No!
Yes!
a(bc)  (ab)c
A( BC )  ( AB)C
Yes!
1 a  a 1  a
IA  AI  A
Real-number multiplication is associative.
Is matrix multiplication commutative?
Real-number multiplication has an identity.
Does matrix multiplication have an identity?
(but you must use an identity matrix of the proper size for A)
Real-number multiplication has inverses.
a  a 1  a 1  a  1
Unless a = 0.
Does matrix multiplication have an identity?
Yes!
AA1  A1 A  I
Unless det(A) = 0.
 2 4
3 2 1 
 1 3
A
and
B




0
4

1


 3 1 
(b) IA
Find: (a) AI
3  2 1 
AI  

0 4  1
A
3  2 1 
IA  

0 4  1
A
(c) BI
 2 4
BI    1 3
 3 1
B
 3 1
1
Show that the inverse of A  
is
A

4
2


1

1  
1 0

3

1



1
2

AA  




3
0
1
2  2


4

2 

 I2
1

 1    3  1
1 0

1
2

A A




3 4
0
1
2


2

2 

 I2
1

 1  2 


3 
2

2 
Matrices: Inverses (and Identities)
Matrix Equations
A system of equations can be written quite simply as AX = C.
To solve it like a real-number equation, we will need the multiplicative
identity and inverse for matrices.
AX  C
A1 AX  A1C
IX  A1C
X  A1C
Multiplicative inverses multiply to the multiplicative identity.
The multiplicative identity disappears from multiplication.
We will have solved the equation.
Matrices: Inverses (and Identities)
Finding The Inverse Of A Matrix
The General Procedure to find the inverse of A.
1. Write a rectangular augmented matrix [ A | I ].
2. Use Gauss-Jordan elimination (row operations on the whole matrix) to
change A into the identity. You will now have [ I | A–1 ].
I can demonstrate this process quickly online here.
If that link doesn’t work right, try this one and use the “row operation calculator.”
A matrix A that has no inverse (because det(A) = 0) is called a singular matrix.
A matrix A that has an inverse (because det(A)  0) is called a nonsingular matrix