Math. 221 (s. 02), Test One, 24 February 2004

Also available: brief answer-sheet to this exam

Scoring: Each of problems 1-5 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 letter-grade 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 insignificant-looking “… and why?” portions of the two parts of question 4 and of part b) of question 5 are just as important as the more impressive-looking 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 row-echelon form, in (fully) reduced row-echelon 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 five-fold products, which of these products will turn out to be zero?  … and which non-zero?

 [ ] 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 one-fifth the sum of the ages of Dick and of Jane.
Dick’s age alone is two-thirds 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  k-by-n  and  n-by-l  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 row-echelon form of  A  has 17 rows of zeros, what (if anything) can be said about how many rows of zeros the fully reduced row-echelon 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?