Some problems

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.

For (1) note that it isn't true for square matrices, so you have to think of non-square A and B.

For (2), note that over Z/2 an antisymmetric matrix is also symmetric and that the sum of symmetric matrices is symmetric.

For (3) write A^2 = (AB)(AB) = A(BA)B = ABB = (AB)B = AB = A.

For (4) consider the degrees.

thank you,
I will try them and if I have any other problems I will let you know.
thank you again