Download 2.1

MAT 2401
Linear Algebra
2.1 Operations with
Matrices
http://myhome.spu.edu/lauw
HW...


If you do not get 9 points or above on
#1, you are not doing the GJE
correctly. Some of you are doing RE.
GJE is the corner stone of this class,
you really need to figure it out.
Today



Written HW
Again, today may be longer. It is
more efficient to bundle together
some materials from 2.2.
Next class session will be shorter.
Preview



Look at the algebraic operations of
matrices
“term-by-term” operations
• Matrix Addition and Subtraction
• Scalar Multiplication
Non-“term-by-term” operations
• Matrix Multiplication
Matrix

If a matrix has m rows and n columns,
then the size (dimension) of the
matrix is said to be mxn.
1
1 

2 


m 

2
n







Notations

Matrix
A   aij 
j th




i th 


aij







Notations

Matrix
Example:
A   aij 
1 1 1 4 
A   2 2 5 11 
 4 6 8 24 
j th




i th 


aij







a11 
a23 
Special Cases

Row Vector
b1

b2
bn 
Column Vector
 c1 
c 
 2
 
 
 cm 
Matrix Addition and Subtraction
Let A = [aij] and B = [bij] be mxn matrices


Sum: A + B = [aij+bij]
Difference: A-B = [aij-bij]
(Term-by term operations)
Example 1
1 2 
A

3

1


 0 2
B


3
2


A B 
A B 
Scalar Multiplication
Let A = [aij] be a mxn matrix and c a
scalar.
Scalar
Product: cA=[caij]
Example 2
1 2 
A

3 1
2A 
Matrix Multiplication


Define multiplications between 2
matrices
Not “term-by-term” operations
Motivation

The LHS of the linear equation consists of
two pieces of information:
• coefficients: 2, -3, and 4
• variables: x, y, and z
2x  3y  4z  5
Motivation

Since both the coefficients and variables
can be represented by vectors with the
same “length”, it make sense to consider
the LHS as a “product” of the
corresponding vectors.
2x  3y  4z  5
 x


 2 3 4  y   5
 z 
Row-Column Product
 a1
a2
 b1 
b 
an   2   a1b1  a2b2 
 
 
bn 
same no. of elements
 anbn
Example 3
2
2
1 3 2 4  1
 
 2 
Matrix Multiplication
 a11


 ai1


 am1









a12
ai 2
am 2
a1 p 
  b11
 b
aip   21

 b

amp   p1
ai1b1 j  ai 2b2 j 
b1 j
b2 j
bpj
 aip bpj
b1n 
b2 n 


bpn 
j th












i th 


cij







Example 4
 1 2  0 1
 1 0   1 0 



Example 5 (a)
 4 2
1 2 1  

0
1
 2 3 1 


 2 1 


Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
Example 5 (b)
1 2 3   2 
 2 3 1  3

 
Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
Example 5 (c)
1
 1 1 2
 
Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
Example 5 (d)
1
1 2  1
 
Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
1
Remark: A  1 2 , B 
 1
 
Example 5 (e)
1 1  1 1 
1 1  1 1



Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
1 1
1 1
, B
Remark: A  


1
1

1

1




Example 5 (f)
1 0  1 2 
 0 1  3 4 



Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
1 0 
1 2
I

,
A

Remark:
0 1 
3 4 




Interesting Facts

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

The product of mxp and pxn matrices is a
mxn matrix.
In general, AB and BA are not the same
even if both products are defined.
AB=0 does not necessary imply A=0 or B=0.
Square matrix with 1 in the diagonal and 0
elsewhere behaves like multiplicative
identity.
Identity Matrix
nxn Square Matrix
1
0
I  In  


0
0
1
0
0


0

1
Zero Matrix
mxn Matrix with all zero entries
0  0mn
0
0



0
0
0
0
0


0

0
Representation of Linear
System by Matrix Multiplication
x yz  4


2 x  2 y  5 z  11
 4 x  6 y  8 z  24










 
 
 

 
 





Representation of Linear
System by Matrix Multiplication
x yz  4


2 x  2 y  5 z  11
 4 x  6 y  8 z  24










 
 
 

 
 





Representation of Linear
System by Matrix Multiplication
x y z  4


2 x  2 y  5 z  11
 4 x  6 y  8 z  24










 
 
 

 
 





Representation of Linear
System by Matrix Multiplication
Let
x yz  4


2 x  2 y  5 z  11
 4 x  6 y  8 z  24










 
 
 

 
 
1 1 1
 x
4
A   2 2 5 , X   y  , b  11 
 4 6 8
 z 
 24





Then the linear system is given by
Remark
It would be nice
if “division” can
be defined such
that:
Let
1 1 1
 x
4
A   2 2 5 , X   y  , b  11 
 4 6 8
 z 
 24
Then the linear system is given by
(2.3) Inverse
HW...


If you do not get 9 points or above on
#1, you are not doing the GJE
correctly. Some of you are doing RE.
GJE is the corner stone of this class,
you really need to figure it out.