Math. 221 (s. 02), Test One, 24 February 2004
Also available: brief answersheet to this exam
Scoring: Each of problems 15 is worth one full point. Partial credit may be earned. Each full point earned will advance you up one further letter grade from E^{–} . Each additional ⅓ point will advance you up one lettergrade notch (obliterating a ^{–} or adding a ^{+} ). Top score possible then becomes an A^{++} .
Time allowed: 40 minutes. (Note that most of these questions take longer to state than to answer.)
Please indicate explicitly, on what you turn
in, that you have respected Wesleyan’s Honor Code.
NB: The insignificantlooking “… and why?” portions of the two parts of question 4 and of part b) of question 5 are just as important as the more impressivelooking questions they follow. Omitting to explain “why” will keep you from earning full credit on questions 4 or 5. (But there’s no “and why” portion to questions 1, 2, or 3; nor to part a) of question 5.)
1. For each matrix below, tell whether it is in rowechelon form, in (fully) reduced rowechelon form, or both, or neither:

[ 





] 

















1 
2 
0 
3 
0 


[ 




] 


[ 




] 


0 
0 
1 
1 
0 


1 
0 
0 
5 







a) 
0 
0 
0 
0 
1 

b) 
0 
0 
1 
3 

c) 
1 
0 
3 
1 


0 
0 
0 
0 
0 


0 
1 
0 
4 


0 
1 
2 
4 




























[ 





] 








[ 




] 


1 
3 
0 
2 
0 


[ 


] 









1 
0 
2 
2 
0 


0 
0 


d) 
1 
7 
5 
5 

e) 
0 
0 
0 
0 
1 

f) 
0 
0 



0 
1 
3 
2 


0 
0 
0 
0 
0 


0 
0 



















2. a) Count up all the inversions in the permutation ( 5 1 4 3 2 ) .
b) When computing the determinant of the following matrix as an alternating sum of 5! (that’s 120) assorted fivefold products, which of these products will turn out to be zero? … and which nonzero?
[ 





] 
1 
2 
3 
4 
5 

2 
0 
0 
0 
0 

0 
0 
0 
1 
0 

0 
0 
3 
0 
0 

0 
5 
0 
0 
0 
c) Using the results of parts a) and b), evaluate the determinant of the matrix just above.
3. If the ages (in years) of Dick, Jane, and Spot
are added together, the sum is 30 (years);
Spot’s age alone is onefifth the sum of the ages of Dick and of Jane.
Dick’s age alone is twothirds the sum of the ages of Jane and of Spot.
Using d , j ,
and s to stand for the ages (in years) of Dick,
Jane and Spot, respectively,
a) Set up a system of simultaneous linear equations representing the information given above; and
b) Using any technique you like (barring intelligent calculators), find Dick’s, Jane’s, and Spot’s ages.
4. Let A and B be arbitrary kbyn and nbyl matrices, respectively.
a) If five of the rows of A are entirely full of zeros, what (if anything) can be said about how many – or which – rows of the product AB are entirely full of zeros? … and why?
b) If the fully reduced rowechelon form of A has 17 rows of zeros, what (if anything) can be said about how many rows of zeros the fully reduced rowechelon form of the product AB has? … and why?
5. a) For square matrices A and B of the same size, should one generally expect to have AB = BA ?
b) What about det(AB) = det(BA) ? … and why?