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 left-right is also in RREF, what can M look like? (Describe all the possibilities, every last possible M.)

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

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 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:

  • (i) adding a multiple of one row to another
  • (ii) rescaling a row by something nonzero
  • (iii) switching two rows
  • a. Show that the bigness of a matrix is not affected by elementary 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.