1. Consider a kxn matrix M full of variables mij, where
i=1..k, j=1..n.
For S a subset of 1..n of size k, let pS
denote the determinant of the kxk submatrix using columns S and all k rows.
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 {pij} are quadratic polynomials in the variables {mi'j'}. Rewrite p14 p23 as a combination of the other {pij}.
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)).
d. Let M be a kxn matrix, and bi denote the bigness of
the left kxi rectangle inside M, for i = 0,1,...,n.
(So b0 is weird to think about; let's call it 0.)
Show that for each i>0, bi+1 - b_i = 0 or 1.
4. We had three kinds of row operations:
b. Let M be a matrix, and (bi) as in 3(d). Show that
bi+1 > bi iff M's reduced row-echelon form
has a pivot in column i+1.
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.)