1. Let A be an mn matrix and B be an nm matrix.

(a) Suppose Ax = 0 has non-trivial solutions.

(i) an BA be invertible?

(ii) Can AB be invertible if m < n?

(iii) Can AB be invertible if m > n?

(b) Suppose Ax = 0 has only the trivial solution.

(i) Must AB be invertible?

(ii) Must BA be invertible?

Justify all your answers. (For each part in (a), if your answer is yes, you have to give an example to support the statement. For each part in (b), if your answer is no, you have to give a counter-example.)