Ma 237
Summer 2013
Homework 3
Carter
In general, the matrix product is defined for a pair of matrices A ∈ M (n, k) and B ∈
M (k, m). The notation indicates that A is a matrix with n rows and k columns while B is
a matrix with k rows and m columns. That is, the number of columns of A is equal to the
number of rows of B.
The first example that we have experienced is the dot product. For example, if
A = [a1 , a2 , . . . , ak ]
and

B=








1

bk

b 

b2 

,
.. 
. 

then

A·B =




[a1 , a2 , . . . , ak ] · 



1

bk

b 

k
X
b2 

1
2
k
a` b ` .
= a1 b + a2 b + · · · + ak b =
.. 

. 
`=1
Recall from our discussion in class that I will be using superscripts to indicate the row
number, while subscripts indicate the column number. In particular, these do not mean
exponents!

4





Even more specifically, Suppose that A = [2, 3, −5] and that B =  −1 . Then


7



4



A · B = [2, 3, −5] ·  −1  = 2(4) + 3(−1) + (−5)7 = −30.


7
1
In this preliminary numeric case, A is a (1 × 3)-matrix, and B is a (3 × 1)-matrix.
In general if A is an (n × k)-matrix and B is a (k × m)-matrix, then the matrix product
A · B is an (n × m)-matrix in which the (i, j)th entry is the dot product of the ith row of
A with the jth column of B. If a`i is the entry in the ith row and `th column of the matrix
A and bj` is the entry in the `th row and jth column of B, then the matrix entry in the ith
row, jth column of the product is
= a`i bj` . Usually, when we first see the formula, it
Pk
`=1
scares us, so let’s see an example.


Let A = 
1 2

−4 2 5

3

1 2 3




and let B =  2 4 6  . Then


3 6 9


A·B =

=
3·1+1·2+2·3
3
1 2
−4 2 5


·


3·2+1·4+2·6
1 2 3



2 4 6 

3 6 9
3·3+1·6+2·9
−4 · 1 + 2 · 2 + 5 · 3 −4 · 2 + 2 · 4 + 5 · 6 −4 · 3 + 2 · 6 + 5 · 9
Compute the matrix product of the following matrices.
1.


2.


3.
0
5.
6.
·
1
1/3 −4/3
0

1
0
 
1 −5/2

0
1

1 −1

0

1/5 0
·
2

0

1
3 4 24

0 1
0

2 −5

0

1

0

·
1
1 −4
2 −5 30
 
1
0
·
·
1


 
 
1/2 0

4.
 
1/3 0
 
·
1
0
1 1 1

0 1 0

5 −4

0

1



=
11 22 33
15 30 45

.
7.

1 −4/5

0
8.

10.


11.






13.
15.
16.

0 1/4

·
 
·
1
0
1
0
0
0
1




0 1
0 0

1

1 0


0 1 0


 

0 1
0
0 0
1




1
0 −1
·  −2 0
0 0 1 
 
0 1 0
0 0
1
0 1 −1
0 0
1
 
 
·
 
3



0 

1
0 −1
 
 
 
0
1 0 −1
 
 
 

0 

0 0 1








0
0 1 0
·  −2 1
0 
 
1
0 0
0
0
3 4 8 24
1 0 0
1 0

3 4 24



 
 
·
 
−2 1 0
1 0 −1
1 0

 
 
·
 


3 4 8 24
 
0 1
0 1
 
0 0
0
0
1 −3
 
1
1/4 −3/4
1
1
0
·
1/3 −4/3 −8/3

0
 
0
0

·
−3 1
1
5 −4 20
 
0

0




14.
1
1/3 −4/3

12.
·
1

9.
 



1 

1
0 −1

0
1
−2 1
0




1
0
17.

1
0




2
−1
−2
1
−1
 
1 1 1 1
 


 

· 2 0 1 4 
0 

 
0 1 1 2
0
18. Solve the system of equations
x
+ y + z = 1
2x
+ z = 4
y + z = 2
Can you relate your solution to the previous several computations?
4