My problem is this:

State whether the following is true or false, provide a reason.

1. Ax= O has only the trivial solution if and only if Ax=bhas a unique solution for every n x 1 column matrixb.

I am leaning towards false. The reason for this lies with a list of equivalent conditions in my book:

If A in an n x n matrix, then the following statements are equivalent:

1. A is invertible

2. Ax=bhas a unique solution for every n x 1 column matrixb3. A

x= O has only the trivial solution

4. A is row equivalent to $\displaystyle I_n$

5. A can be written as a product of elementary matrices.

The reason I am leaning towards false is because the problem makes it seem that part one of it (Ax= O...) can only be true if part two (Ax = b...) is. Perhaps I am misinterpreting however. Thanks for input.