If a is a negative number, then a(1/a) is equal to -1. if a is a negative number and b is a negative number, then product ab is equal to a negative number

Both claims are false. In fact, and are one the multiplicative inverse of the other. This means, by definition of multiplicative inverse, that

So, it doesn't matter if is positive or negative: the multiplication of one number and its inverse will always be 1: for example,

Similarly, when you multiply two number, the sign of the product depends on the sign of the factors as follows:

So, the multiplication of two negative numbers is a positive number.