/Solutions

1. Diagonal matrices

A matrix A is a diagonal matrix if it is a square matrix with A_ij=0 whenever i≠j.

1. Prove or disprove: If A and B are diagonal matrices of the same size, so is AB.

2. 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

1. Show that there exists a matrix A such that A≠0 but A²=0.
2. 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.

1. Prove that if Ax=0 for some column vector x≠0, then A^-1 does not exist.

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

3. Prove that if the rows of A are not linearly independent, then A^-1 does not exit.