Math 45 — Linear Algebra
September 28, 2016
Name: Answer Key
David Arnold
Quiz #4
Instructions. (15 points) This quiz is open notes, open book. This includes any supplementary texts or
online documents. You must answer all of the exercises on your own. You are not allowed to work in groups
or pairs on the quiz. You are not allowed to enlist the aid of a tutor, classmate, or friend to help with the
quiz. All work must be your own. Place your solution to each problem in the space provided.
Honor Pledge: I promise that the solution submitted was done entirely by me. I received no help from
colleagues, friends, or tutors. I also did not share any parts of my solution with any of my classmates.
Signature:
Staple this quiz cover atop your pencil and paper calculations.
(10pts )
1. Given
1
A=
2
2
0
−1
1

1
B = −1
2
and

0
2
3
perform each of the following tasks:
(a) Using pencil and paper calculations, calculate (AB)T . Show each of your steps.
Solution: We calculate (AB)T .

1 2
(AB)T = 
2 0
−3
4
−3
=
1
=

1
−1 
−1
1
2
T
0
2
3
T
1
3
4
3
(b) Using pencil and paper calculations, calculate B T AT . Show each of your steps.
Solution: We calculate B T AT .
T
T
1 0 1 2 −1
T T


B A = −1 2
2 0 1
2 3


1 2
1 −1 2 
2 0
=
0 2 3
−1 1
−3 4
=
1 3

Therefore, (AB)T = B T AT .
(c) Use Mathematica’s Transpose command to perform each of the calculations in part (a) and part (b).
Obtain a printout of your notebook and staple it to the bottom of your quiz.
Solution: See: quiz4notebook.nb
Math 45 — Linear Algebra/Quiz #4
(5pts )
– Page 2 of 2 –
Name: Answer Key
2. Given that A and B are square matrices, prove that AB = BA if and only if A2 − B 2 = (A + B)(A − B).
Note that this means that you have two “if then” statements to prove. You must show all of your steps,
convincing your instructor of what you are doing at each step.
Solution: Because we have an if and only if statement, we have two things we must prove.
1. If AB = BA, then A2 − B 2 = (A + B)(A − B).
2. If A2 − B 2 = (A + b)(A − B), then AB = BA. We start with number (1).
Proof of #1: Assume that AB = BA. Then:
(A + B)(A − B) = A2 − AB + BA − B 2
And because AB = BA, we can continue as follows:
= A2 − AB + AB − B 2
= A2 + 0 − B 2
= A2 − B 2
Proof of #2: Assume that A2 − B 2 = (A + B)(A − B). Then:
A2 − B 2 = (A + B)(A − B)
A2 − B 2 = A2 − AB + BA − B 2
Now we can subtract A2 from both sides of the equation and add B 2 to both sides of the equation.
0 = −AB + BA
Now we can add AB to both sides of the equation.
AB = BA
Because we have proved both “if, then” statements, we have shown that it is true that AB = BA
if and only if A2 − B 2 = (A + B)(A − B).