Lecture 7: Properties of Matrix Multiplication
Winfried Just
Department of Mathematics, Ohio University
January 25, 2016
Winfried Just, Ohio University
MATH3200, Lecture 7: Properties of Matrix Multiplication
Some familiar-looking properties
Let A = [aij ]k×m , B = [bij ]n×p , and C = [cij ]q×r be matrices. Then
A(BC) = (AB)C = ABC
(Associativity Law),
We need here: m = n and p = q.
The order of ABC will then be k × r .
A(B + C) = AB + AC
(Left Distributivity Law),
We need here: m = n = q and p = r .
The order of A(B + C) will then be k × r .
(B + C)A = BA + CA
(Right Distributivity Law),
We need here: k = p = r and n = q.
The order of (B + C)A will then be q × m.
Homework 15: Prove the Left Distributivity Law.
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Applications of the Associativity Law
We have used the Associativity Law A(BC) = (AB)C already
when we considered compositions of transformations:
TA (TB (~v)) = A(B~v) = (AB)~v = (TA ◦ TB )(~v).
Our next application of the Associativity Law will rely on the
observation that sums of vectors can be expressed as inner
products:
1 1 ...
 
x1
" n
#

X
x

 2
1 .=
xn = x1 x2 . . .
 .. 
`=1
xn
Ohio University – Since 1804
Winfried Just, Ohio University
 
1


1
xn  . 
 .. 
1
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Calculating grades: The instructor’s spreadsheet again

Let
a11
 a21

A= .
 ..
a12
a22
..
.
...
...

a1n
a2n 

..  = [aij ]m×n
. 
am1
am2
...
amn
Assume m is the number of students in the class, n is the number
of gradable items, and the entry aij represents the score of
student i on gradable item number j.
Let ~x be the vector of total scores for each student and let ~y be a
vector of mean scores for each item.
Question: (a) How can you express ~x as the product of A and a
vector ~v? Would the result of this operation give you a row vector
or a column vector ~x?
~?
(b) How can you express ~y as the product of A and a vector w
Would the result of this operation give you a row vector or a
column vector ~y?
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
The answer:

Let
a11
 a21

A= .
 ..
a12
a22
..
.
...
...

a1n
a2n 

..  = [aij ]m×n
. 
am1
am2
...
amn
Assume m is the number of students, n is the number of gradable
items, and aij the score of student i on gradable item number j.
Let ~x be the vector of total scores for each student and let ~y be a
vector of mean scores for each item.
 
1
1
 
Then ~x = A  .  and ~y = m1 m1 . . . m1 A
 .. 
1
The row vector has m entries and the column vector has n entries.
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Another use of the Associativity Law: Homework 1 again
Let A be the instructor’s spreadsheet.
Method 1 of calculating the mean total score, adding up the total
scores of all students and dividing the result by m, can be
expressed as the inner product
  
1
  . 
1 1 . . . 1 A  .. 
m
1
Similarly, Method 2 of calculating the mean total score, adding up
the mean scores of all gradable items, can be expressed as
 
 
1
1
1
.
.
1 1
.
1 . . . 1 A  .. 
m ...
m A . = m
1
1
The Associativity Law implies that the two products are equal.
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Commutativity fails
Even for square matrices A = [aij ]n×n and B = [bij ]n×n we may
have AB 6= BA. Consider the example:
0 1 4 5
c11 c12
AB =
=
2 3 6 7
c21 c22
4 5 0 1
d11 d12
BA =
=
6 7 2 3
d21 d22
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Commutativity fails
Even for square matrices A = [aij ]n×n and B = [bij ]n×n we may
have AB 6= BA. Consider the example:
0 1 4 5
6 c12
AB =
=
2 3 6 7
c21 c22
c11 = a11 b11 + a12 b21 = 0 + 6 = 6.
4 5 0 1
10 d12
BA =
=
6 7 2 3
d21 d22
d11 = b11 a11 + b12 a21 = 0 + 10 = 10.
Since c11 6= d11 , the matrices AB and BA differ in at least one
element and we conclude that AB 6= BA.
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
The transpose of a product
Let A = [aij ]k×n and B = [bij ]n×p .
Is then (AB)T = AT BT ?
Not in general.
The expression on the right may not even be defined.
But we always have: (AB)T = BT AT
Since BT has order p × n and AT has order n × k,
the product BT AT is defined.
(We are assuming here that all elements of these matrices are
numbers, which we will from now on always do when we work with
the transpose.)
Homework 16: Test the above claims for
1
A=
B= 3 4 5
2
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication
Multiplication of symmetric matrices is commutative
Let A = [aij ]n×n and B = [bij ]n×n be symmetric,
that is, AT = A and BT = B.
Homework 17: Assume A and B are symmetric and of the same
order. Prove that the product AB is also a symmetric matrix.
That is, prove the (AB)T = AB.
Now let A, B be symmetric. Then:
BA = BT AT = (AB)T = AB.
The first equality follows from the definition of symmetry.
The second equality follows from reading (AB)T = BT AT
backward.
The third equality follows from the result of Homework 17.
Thus multiplication of symmetric matrices is commutative.
Ohio University – Since 1804
Winfried Just, Ohio University
Department of Mathematics
MATH3200, Lecture 7: Properties of Matrix Multiplication