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Lecture 4

MTH 341 Lecture Notes - Lecture 4: Cross Product, Branwen, Zero MatrixExam


Department
Mathematics
Course Code
MTH 341
Professor
Rockwell, Dan
Lecture
4

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Math 341 - Wednesday, April 10th, 2019 Page 1 of 3
Name:
Directions: Discuss and solve the following problems in your groups. You are encouraged
to write out your work on additional pieces of paper.
(1) True or False:IftwomatricesAand Bhave the same reduced row-echelon form,
then A=B. (Write out an explanation for your answer or provide an example.)
(2) True or False:IfthesumA+Bis defined, then the product AB is also defined.
(Write out an explanation for your answer or provide an example.)
(3) Let A=2
4
31 2
721
10 2
3
5and B=2
4
421
24 3
512
3
5.
Find the following if possible.
(a) 3A2B
(b) BA
(4) Write the following systems of linear equations in “vector form”.
(a) 8
<
:
x+y+z=0
2xy+z=1
x+yz=2
(b) 8
<
:
xy+z=2
2x+3y+z=1
2x2y+2z=4
(c) 8
<
:
xy+z+w=2
x3y+2z4w=12
x3y+2z+2w=17
(d) 8
>
>
<
>
>
:
x1+x2+x3x4=0
x1x2+x3x4=0
x1+x2x3x4=0
x1+x2+x3x4=0
(5) For the systems in problem 4 rewrite them in matrix-vector form. (Our book just
calls this matrix form.)
(6) Note: The connection between the “vector form” and the “matrix-vector” form of a
system is a helpful way to understand matrix vector products. When we talk about
column spaces and the image of a matrix the “vector” form of a matrix vector product
will be useful for our understanding.
Task: When your group gets to this item check in with Dan, Branwen, or Nadia to
look over your various forms.

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Math 341 - Wednesday, April 10th, 2019 Page 2 of 3
There are various ways that matrix multiplication can be defined. Section 2.1.3 de-
fines matrix multiplication in way that is useful for conceptual understanding. Below
is a definition from section 2.1.4 that defines matrix multiplication in a way that is
useful for computation.
Definition: (Dot Product of two vectors) Let ~w=2
6
6
4
w1
w2
...
wn
3
7
7
5
and ~v=2
6
6
4
v1
v2
...
vn
3
7
7
5
then the
dot product of ~wand ~vis denoted ~w·~vand is equal to the scalar defined by
n
X
i=1
wivi.
(7) Let ~p=3
1and ~n=4
2. Find ~p·~n.
Definition: (Matrix Multiplication)
Let Abe a (m×n)matrix(e.g.Ais a matrix with mrows that each have nentries)
and Bbe a (n×p)matrix(e.g.Bis a matrix with pcolumns with nentries).
Let Aibe the ith row of Aand let Bjbe the jth column of B.
The the product of Aand B,denotedAB,isan(m×p)matrixandisdenedentry-
wise with its i, jth entry equal to
ci,j =Ai·Bj.
(This definition is similar to what you will find on page 65 of our textbook.)
(8) Let C=31
72
and D=42
24
.
Find the following if possible.
(a) CD
(b) DC
(9) Let A=2
4
31 2
721
10 2
3
5and B=2
4
421
24 3
512
3
5.
Find the following if possible.
(a) AB
(b) BA

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Math 341 - Wednesday, April 10th, 2019 Page 3 of 3
For the following problems consider the matrices below.
A=2
6
6
4
321
443
917
7212
3
7
7
5
,B=2
6
6
4
10
01
11
02
3
7
7
5
,C=2
4
424
212
424
3
5,
D=101
20 0
,E=2
4
12 1
42
53
3
5,F=2
4
0111
10 11
11 01
3
5
G=2
4
301
01 3
002
3
5,H=211
01 5
,J=2
6
6
4
831
400
902
010
3
7
7
5
(10) Make a list of all of the products that are defined given the matrices above.
(11) Make a list of all of the sums that are defined given the matrices above.
(12) Compute the products in your list from problem 10.
(13) Compute the sums in your list from problem 11.
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