Math. 221, Key to Test One, 24 February 2004
1. For each matrix below, tell whether it is in row-echelon form, in (fully) reduced row-echelon form, or both, or neither:
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a): both; b): neither (until you switch rows 2 and 3); c): both; d): just in r-e form; e): neither; f): both.
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2. a) Count up all the inversions in the permutation ( 5 1 4 3 2 ) .
Four
inversions after 5; none after 1; two after 4; one
after 3. Total, all told: seven inversions.
b) When computing the determinant of the following matrix as an alternating sum of 5! (thats 120) assorted five-fold products, which of these products will turn out to be zero? and which non-zero?
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All 24 products using the 1 from row 1 will have to
pick up a factor 0 from row 2; all 24 using the 2 from row 1 will have to pick
up a zero from row 5; those using the 3 from row 1 will have to pick up a zero
from row 4; and those using the 4 from row 1 will have to pick up a zero from
row 3.
So only the 24 products using the 5 from row one have
any chance of being non-zero, and, of those,
all pick up a zero from some row, somewhere along the line, except for
the one using the 5 from row 1, the 2 from row 2, the 1 from row 3, the 3 from
row 4, and the 5 from row 5. That product, then, which is the product arising
from the permutation of part a), is (5)(2)(1)(3)(5) = 150 (all 119 others =
0),
c) Using the results of parts a) and b), evaluate the determinant of the matrix just above.
and so the determinant in question is (1)7(150) = 150 .
3. If the ages (in years) of Dick, Jane, and Spot
are added together, the sum is 30 (years);
Spots age alone is one-fifth the sum of the ages of Dick and of Jane.
Dicks 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;
One possible system is: d + j + s = 30 ; d + j 5s
= 0 ; 3d 2j
2s = 0 .
b) Using any technique you like (barring intelligent calculators), find Dicks, Janes, and Spots ages.
To make a long story short, Dick,
Jane, and Spot are 12, 13, and 5 years old, respectively.
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? If the i th row of A is all-zero then the i th row of AB will be all-zero, too (see why?). Thus AB will have rows of zeros at least wherever A does. In particular, AB will have at least five all-zero rows, in the same places A does.
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? The frref of A can be written EA for suitable invertible E. If EA ends with 17 rows of zeros, so will (EA)B = E(AB) . But the frref of AB coincides with the frref of E(AB), and the frref of E(AB) cant have any fewer all-zero rows than has EAB itself. Thus frref(AB) also has at least 17 rows of zeros.
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?
Part a): No! Part b): Yes! Indeed, det(AB) = det(A)det(B)
= det(B)det(A) = det(BA) .