Ma 237-101(CRN:30074) Summer 2015 HW 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


1
 b 
 2 
 b 


,
B=
 .. 
 . 

bk

then


1
 b 
 2 
k
 b 
X


A · B = [a1 , a2 , . . . , ak ] ·  .  = a1 b1 + a2 b2 + · · · + ak bk =
a` b ` .
 . 
 . 
`=1

bk

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!
1

4





Even more specifically, Suppose that A = [2, 3, −5] and that B =  −1 . Then


7


4



 −1  = 2(4) + 3(−1) + (−5)7 = −30.
A · B = [2, 3, −5] · 


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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.
1/3 0
0
1
1/3 −4/3
0


1
1/2 0
0
1
2
 
·
 
·
 
·
1 −4

0

1
3 4 24

0 1
0

2 −5

0

1



=
11 22 33
15 30 45

.
4.

1 −5/2

0
1

1 −1
5.

1/5 0
1 −4/5

0
8.

9.


11.






13.
·
1

0 1/4
0

1
0
0
0
1
0 0
1

0

1

1 0

1
1 −3


3 4 24

1 0

0
3 4 8 24

0 1 0

0
 
3 4 8 24

 
 
·
 
0 1 0


 
 
 
0 0 1
1 0 −1
· 0 1
−2 1 0 
 
0 0 1
0 0
3
0
0 1
0
·
·
0





5 −4 20
 
1
1
5 −4
 
1
1/3 −4/3 −8/3

0 1 0

·
1/4 −3/4
0

 
0
0
1 1 1
·
−3 1

0
 
0
1/3 −4/3

12.
1

1
·
 
1

0
 
1


·
1
0
2 −5 30
 
0

10.
·

6.
7.
 



0 

1
0 

0
14.
15.
16.
17.

1 0 −1




0 1
 
1
0 −1
 
 
·  −2 1
0 
 
0 0
1
0 0
 

1 0 0




·  −2 0
0 0 1 
 
0 0
0 1 0
1
 

1 0




· 0 0
0 1 −1 
 
−2 1
0 0 1
0
1
 
 

1
0




2
−1
−2
1
−1
 


0 

1
0 −1
 
 




1 

1
0 −1



1 

0
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?
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