epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Review Exercises 105
19. Let y4 be a 2 X 2 matrix with real entries. The following statements are
equivalent: (*)
(1) The matrix A has an inverse.
(2) The determinant of A is non equal to zero.
(3) The system AI J = I J has only the trivial solution x = 0, y = 0.
20. Let/(x) = 3x2 ^ 7^ jy^^^ ^^^ y^^) ^ iQ
21. For all positive integer numbers k:
22. Let a and b be two real number. If ab = ^^^^, then a = b.
23. Let a and b be two real number. If ab = ^^^^, then a = b = 0.
24. For all integers /c > 2,
1 1 1 1
fc+l k-\-2 "" 2k T
25. Let fl, fe, and c be three integers. If a is a multiple of b and ft is a multiple
of c, then a is a multiple of c.
26. Let p be a nonzero real number. Then /? is rational if and only if its
reciprocal is a rational number.
27. Let W = (-1/2)"}^,. Then lim a„ = 0.
I 3n—i n-^00
28. Let a, b, and c be three consecutive integers. Then 3 divides a + fc + c.
29. Let fc be a whole number. Then fe^ — feis divisible by 3. How does this
exercise relate to the previous exercise?
30. Let {a„}^i be a sequence of real numbers converging to the number L;
that is lim an = L
«->00
If a„ > 0 for all n, then L > 0.
31. Let/(x) = V^. Then lim/(x) = \/3.
x->3
32. If ad — bc^ 0, then the system:
ax-\-by = e
cx-^dy =f
has a unique solution. The numbers a, b, c, d, e, and / are all real
numbers.
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