Some Viewpoints on Matrix Multiplication.
a b
e f
Let A =
and B =
. Then
c d
g h
a b
e f
ae + bg
AB =
=
c d
g h
ce + dg
af + bh
cf + dh
There are many ways to interpret this remarkable operation. Perhaps the most natural
one being the interpretation of matrix multiplication as function composition (of linear
functions) from which the associativity follows directly. But the following formulas suggest
other interpretations.

a
b

= e
+g
c
d

a
b 
f
+h
c
d
i.e. multiplication of A on the right by a matrix corresponds to column operations on A.


a[e f ] + b[g
=
c[e f ] + d[g
h]

h]
i.e. multiplication of B on the left by a matrix corresponds to row operations on B.
ae af
bg bh
a
b
=
+
=
[e f ] +
[g h]
ce cf
dg dh
c
d
i.e. we can express AB as a sum of matrices each of rank 1.
In Matrix Algebra, you can choose which interpretation you wish to use at a given
point in an argument.
What follows
are the
for3 × 3 matrices.

 same calculations

a b c
q r s
Let A =  d e f  and B =  t u v . Then
g h i
x y z

a b

AB = d e
g h

c
q r


f
t u
i
x y
 
s
aq + bt + cx ar + bu + cy


v = dq + et + f x dr + eu + f y
z
gq + ht + ix gr + hu + iy


as + bv + cz
ds + ev + f z 
gs + hv + iz

 
 
 
 
 
 
 
 
 
a
b
c
a
b
c
a
b
c 

AB = 


















q
d
+
t
e
+
x
f
r
d
+
u
e
+
y
f
s
d
+
v
e
+
z
f


g
h
i
g
h
i
g
h
i
1
i.e. multiplication of A on the right by a matrix corresponds to column operations on A.


 a[q r s] + b[t u v] + c[x y z ] 


AB =  d [ q r s ] + e [ t u v ] + f [ x y z ] 


g[q r s] + h[t u v] + i[x y z]
i.e. multiplication of B on the left by a matrix corresponds to row operations on B.

aq

AB = dq
gq
 
a
= d[q
g
ar
dr
gr
r
 
bt
as


ds + et
gs
ht
bu
eu
hu
 
b
s] +  e [t
h
u
 
bv
cx cy


ev + f x f y
hv
ix iy
 
c
v] + f [x y
i
i.e. we can express AB as a sum of matrices each of rank 1.
2

cz
fz 
iz
z]