I have a ton of homework, but I am stuck on 4 of them.

1. Find non-invertible matrices A and B, such that AB is invertible.

2. Find, with proof, a square matrix A with entries in Z2 such A is not the sum of a symmetric and anti-symmetric matrix.

3. A square matrix X is called idempotent if X^2 = X. Prove that if AB = A and BA = B then A and B are idempotent.

4. Let Pn(x) be the vector space of all polynomials of degree <= n with real number coefficients. Prove that the set

{1, x, x(x − 1), x(x − 1)(x − 2), . . . , x(x − 1)(x − 2) • • • (x − n + 1)}

is a basis for this vector space.

If anybody can help me with any I will appreciate it.