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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
(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.
(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.

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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

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

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.