Math240, Spring 2007
Answer Key
1. (i) false, (ii) true, (iii) false.
2. Solution is not unique, a possible solution is as follows. Since
det(AT) = det(A), we obtain that det(A) = det(−A) = (−1)3det(A) =
−det(A). So, det(A) = 0, and therefore Ais singular and its rank is
smaller than 3.
3. Answer is not unique, a possible answer is
P=
1 1 1
1−1 0
1 0 −1
, D =
0 0 0
0−3 0
0 0 −3
.
4. (i) true, (ii) true, (iii) true.
5. c2= 0 and cn+3 =2
(n+3)(n+2) cnfor n≥0; a possible choice of y’s
is as follows: y1= 1 + 1
3x3+1
45 x6+. . . and y2=x+1
6x4+1
126 x7+. . .
(a more tricky choice is to take, for example, y1= 1 + x+1
3x3+. . .
and y2= 1 −x+1
3x3+. . . ).
6. y1= 10e5t+ 6e−t, y2= 5e5t−6e−t.
7. y= (11e−3t+ 3 sin(t)−cos(t)) /10.
8. (g)
9. (f)
10. (g)
11. (c)
12. (c)
13. (b)
14. (h)
15. (e)
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