For S a subset of 1..n of size k, let p

a. If k=2, how many different such subsets S are there, as a function of n?

b. Say k=2 and n=4, so the subsets are {12,13,14,23,24,34}. Then the
{p_{ij}} are quadratic polynomials in the variables {m_{i'j'}}.
Rewrite p_{14} p_{23} as a combination of the other
{p_{ij}}.

2. Let M be a matrix in reduced row-echelon form.

a. If M flipped upside-down is in RREF (but not left-right),
what can M look like?
(Again, describe

3. Define the **bigness** of a matrix M as the size r of the
largest rxr submatrix with nonzero determinant.
By "submatrix" I mean, you cross out a bunch of rows and columns,
no assumptions on the remaining rows or columns being consecutive.

a. What does a matrix of bigness 0 look like?

b. Say M = c r, where c is a column vector and r a row vector. Show that M has bigness 0 or 1.

c. Say M has bigness exactly 1 (so, not like part (a)).

Show that M can be written as a column vector times a row vector.
(Meaning, find a c and an r that do the job.)

d. Let M be a kxn matrix, and b_{i} denote the bigness of
the left kxi rectangle inside M, for i = 0,1,...,n.
(So b_{0} is weird to think about; let's call it 0.)
Show that for each i>0, b_{i+1} - b_i = 0 or 1.

4. We had three kinds of row operations:

a. Show that the bigness of a matrix is not affected by elementary row operations.
b. Let M be a matrix, and (b_{i}) as in 3(d). Show that
b_{i+1} > b_{i} iff M's reduced row-echelon form
has a pivot in column i+1.