I need to know what you've covered and what tools are available. The proof of the invertible matrix theorem is widely available all over Google so I suggest skimming through some of those proofs and then posting any followup ideas or questions.
Many of these proofs also work by proving a couple of statements and then using that to imply the other statements. Any true statement of the IMT implies all of the others so there are lots of ways to go between these ideas.
Here is an example of an answer to your question:
"Assume that for the matrix A, Row i = Row j. By interchanging these two rows, the determinant changes sign (by Property 2). However, since these two rows are the same, interchanging them obviously leaves the matrix and, therefore, the determinant unchanged. Since 0 is the only number which equals its own opposite, det A = 0"
This uses the property that switching two rows of a matrix will reverse the sign of the determinant.