#17

Let * be the binary operation on the rational numbers given by a * b = a + b + 2ab, which of the following are true?

I: * is commutative

II: There is a rational number that is a *-indentity

III: Every rational number has a *-inverse

The answer is C: I and II.

Obviously, * is commuatative, but I have problem finding the indentity. Can anyone show me?

#18

A group G in which (ab)^2 and a^2 b^2 for all a, b in G is necessarily

(A) finite

(B) cyclic

(C) of order two

(D) abelian

(E) none of the above.

The answer is D.

Can anyone explain this to me?

#35

The rank of the matrix is?

| 1 2 3 4 5 |

| 6 7 8 9 10|

| 11 12 13 14 15|

| 16 17 18 19 20|

| 21 22 23 24 25|

The answer is 2.

Can anyone show me?