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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

72−1

10 2

3

5and B=2

4

42−1

24 3

51−2

3

5.

Find the following if possible.

(a) 3A−2B

(b) B−A

(4) Write the following systems of linear equations in “vector form”.

(a) 8

<

:

x+y+z=0

2x−y+z=1

x+y−z=2

(b) 8

<

:

x−y+z=−2

2x+3y+z=1

2x−2y+2z=−4

(c) 8

<

:

x−y+z+w=2

x−3y+2z−4w=−12

x−3y+2z+2w=17

(d) 8

>

>

<

>

>

:

x1+x2+x3−x4=0

x1−x2+x3−x4=0

x1+x2−x3−x4=0

−x1+x2+x3−x4=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)matrixandisdeﬁnedentry-

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

72−1

10 2

3

5and B=2

4

42−1

24 3

51−2

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

−32−1

44−3

917

7−212

3

7

7

5

,B=2

6

6

4

−10

01

1−1

02

3

7

7

5

,C=2

4

424

212

424

3

5,

D=10−1

20 0

,E=2

4

12 1

4−2

−53

3

5,F=2

4

01−11

10 11

11 01

3

5

G=2

4

−30−1

01 3

00−2

3

5,H=−21−1

01 5

,J=2

6

6

4

8−31

−400

90−2

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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