# 1. Diagonal matrices

A matrix A is a **diagonal matrix** if it is a square matrix with A_{ij}=0 whenever i≠j.

- Prove or disprove: If A and B are diagonal matrices of the same size, so is AB.
Let p(A)=Π

_{i}A_{ii}. Prove or disprove: If A and B are diagonal matrices as above, then p(AB) = p(A)p(B).

# 2. Matrix square roots

- Show that there exists a matrix A such that A≠0 but A²=0.
- Show that if A²=0, there exists a matrix B such that B²=I+A. Hint: What is (I+A)²?

# 3. Dimension reduction

Let A be an n×m **random matrix** obtained by setting each entry A_{ij} independently to ±1 with equal probability.

Let x be an arbitrary vector of dimension m.

Compute E[||Ax||²], as a function of ||x||, n, and m, where ||x|| = (x⋅x)^{1/2} is the usual Euclidean length.

# 4. Non-invertible matrices

Let A be a square matrix.

Prove that if Ax=0 for some column vector x≠0, then A

^{-1}does not exist.Prove that if the columns of A are not linearly independent, then A

^{-1}does not exist.Prove that if the rows of A are not linearly independent, then A

^{-1}does not exit.