2.1 Matrix Operations
Notation: An m x n matrix A can be written in
two ways:
1. In terms of the columns of A:
a1
a2  an 
2. In terms of the entries of A:
 a11
 

A   ai1

 
 am1




a1 j

aij

amj



a1n 
 
ain 

 
amn 
The diagonal entries are a11, a22 , a33... .
The zero matrix is
0  0  0






0  0  0  0





 0  0  0 
1
Theorem 1: Let A, B, and C be matrices of the
same size and let r and s be scalars. Then:
a. A  B  B  A
b. A  B   C  A  B  C 
c. A  0  A
d .r  A  B   rA  rB
e.r  s A  rA  sA
f .r sA  rs A
Matrix Multiplication:
Recall: multiplying B and x results in the
vector Bx.
Multiplying A times Bx transforms Bx into the
vector A(Bx). So, A(Bx) is the composition of
two mappings:
x → Bx → A(Bx).
When we define the product AB, we define it
so that A(Bx) = (AB)x.
2
Suppose A is m x n and B is n x p where


B  b1 b 2  b p and
 x1 
 
x  
x p 
 
Then,
Bx  x1b1  x2b 2    x pb p
and
ABx   Ax1b1  x2b 2    x pb p 
 Ax1b1  Ax2b 2    Ax pb p
 x1 Ab1  x2 Ab 2    x p Ab p

 Ab1 Ab 2
 x1 
 
 Ab p  x2 
xp 
 



Therefore, ABx   Ab1 Ab 2  Ab p x
We define
AB  Ab1 Ab 2  Ab p , so A(Bx) = (AB)x.
3
Example: Compute AB where
 4  2
 2  3
A  3  5
B

&
6

7


0 1 
AB  Ab1 Ab2 
  4  2
2


  3  5  6 
 0 1   
42    2 6
 32    56 


 02   16 

4  2
3  5    3 

  7  
0 1    
4 3   2   7 
 3 3   5  7 


 0 3  1  7 
 -4 
- 24


 6 
2
 26 
 
 7 
 -4 2 
- 24 26 


 6  7 
4
Note: Ab1 is a linear combo of the columns of
A, and Ab2 is a linear combo of the columns
of A.
►Each column of AB is a linear combo of the
columns of A using weights from the
corresponding columns of B.
The definition is more important than being
able to multiply matrices by hand.
Example: Suppose A is 4 x 3 and B is 3 x 2.
What is the size of AB?
What is the size of BA?
In general, if A is m x n and B is n x p, then
AB is m x p.
An Alternate method for Computing AB
Let
2  3
 2 3 6
0 1 
A
B





1
0
1


4 7 
5
 22  30  64 2 3  01  67  
 12  00  14  1 3  01  17 


28 36
 2 10 


Note: BA is 3 x 3. Multiplication is not
commutative, but matrices have some of the
properties of real numbers.
Theorem 2
If A is an m x n matrix and B and C have
sizes for which the indicated sums and
products are defined,
a. (AB)C = A(BC)
b. A(B + C) = AB + AC
c. (B + C)A = BA + CA
d. r(AB)= (rA)B = A(rB) for any scalar r.
e. ImA = A = AIn
6
WARNINGS
These properties are analogous to those we
have for real numbers, but not all properties
of real numbers apply to matrices.
1. No commutativity
2. If AB = AC, then B may not equal C.
3. It is possible to have AB = 0 and A ≠ 0
and B ≠ 0.
Powers of A
A 
A

A
k
k times
Example:
3
1 0 1
3 2  3

 
1 0 1

 3
9
4


0 1 0 1 0
2 3 2 3 2
0
2
 1 0


21
8


7
Transposes
If A is an m x n matrix, the transpose of A is
the n x m matrix, denoted by AT, whose
columns are formed by the corresponding
rows of A.
Example:
1 2 3 4 5
A  6 7 8 9 8
7 6 5 4 3
1
2

T
A  3

4
5
6
7
8
9
8
7
6
5

4
3
Example:
 1 2
1 2 0
 0 1
A
B




3
0
1


 2 4
Compute: AB, (AB)T, ATB T, BTA T
8
1 4 
1 1 
T
 AB  
AB  


1
10
4
10




1 3
1 0  2 


T T
A B   2 0 

2
1
4

0 1 
7 3 10 


  2 0  4
2 1 4 
1 3
1 0  2  

T T
B A 
2 0


 2 1 4  0 1 


1 1 


4
10


We notice that (AB)T = BTA T. This is not a
coincidence.
9
Theorem 3: Let A and B denote matrices
whose sizes are appropriate for the following
sums and products.
 
T T
a. A
A
b. A  B   AT  BT
T
 
c.rA   r AT for any scalar r
T
d . AB   BT AT
T
10