Matrices, Equivalent Conditions
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 = b has a unique solution for every n x 1 column matrix b.
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 = b has a unique solution for every n x 1 column matrix b
3. Ax = O has only the trivial solution
4. A is row equivalent to
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.