# Math Help - Prove (or disprove) nonsingular in these two statements?

1. ## Prove (or disprove) nonsingular in these two statements?

A = n X n matrix, I is the identity matrix

1) If A ^ 3 = O, then A - I is nonsingular

2) If the product of k matrices A1 ... Ak is nonsingular, then each matrix Ai is non singular.

Any thoughts?

2. For # 1, I would take the equation

$A^{3}=0$ and subtract $I$ from both sides.

For # 2, I would go with determinants.

How do these ideas work for you?

3. Thanks, I can do it with the methods you suggested. However, the determinant section in my book comes after the section with these problems, so I'm kinda unsure if this method will be accepted. But thanks anyway.

4. Well, for # 2, you could work with the row-reduced (upper triangular) versions of each matrix, and then focus on the last row-last column entry. You could do the same with # 1, after factoring.