Homework #4 for Math H110, fall '02
<p>
1. Let V be (F_2)^2, the plane with coefficients from F_2 = {0,1}, 1+1=0.
Find a bilinear function NOTDET: V x V -> F with the property that
NOTDET(a,b) = -NOTDET(b,a) where a,b from V. (Of course that number is
also +NOTDET(b,a), since 1=-1.) Also, I demand that this NOTDET not be a 
multiple of the determinant (thinking of a,b as the columns of a 2x2 matrix.)
(Of course, there are only two multiples.)

<p>
A. Let A = NOTDET([1,0],[1,0]) 
B = NOTDET([1,0],[0,1]) = NOTDET([0,1],[1,0]) equal because of the
(anti)symmetry, and C = NOTDET([0,1],[0,1]). 
By bilinearity of NOTDET, we find out that 
<center>
NOTDET([x,y], [v,w]) 
= A xv + B (xw + yv) + C yw
</center>
To get det, we use A=C=0, B=1. To get the other
multiple of det, we use A=B=C=0. Anything else gives a nonmultiple.
Perhaps the simplest is NOTDET([x,y], [v,w]) := xv.

<p>
2.
Call a square matrix U a <b>positive root matrix</b> if it's
the identity matrix plus one extra entry above the diagonal.
Figure out what left or right multiplication of a matrix M by a root
matrix U does to M. (Not a hard question.) Show also that det M = det UM
= det MU.


<p>
A. 
Left multiplication adds a multiple of a row to a higher row.
Right mult adds a column to a column further right. 
Using linearity, and repeated-row-death (include det T = det T^* to get
repeated-column-death too), we get the statement about the dets.
<p>

3.
Define a <b>monomial matrix</b> to be square, with at most one nonzero entry
in each row and column. Show that every square matrix M can be reduced to a 
monomial matrix by left and multiplication by positive root matrices.
(If you have trouble, start with the 2x2 case.) 
Put another way, every matrix 
is a product of positive root matrices, a monomial matrix, 
and some more positive root matrices. 
<p>
A. Look at the lowest nonzero entry in the left column (or second column
if the first column is zero, or third column, etc.). Use upward-row 
operations to
cancel out all the other entries in that column. Use rightward-column
operations to cancel out all the other entries in that row.
<p>
Now mentally throw out that row and column and don't do any further
operations that involve them. By induction we can assume that the
smaller (n-1)x(n-1) matrix can be "monomialized" (not standard terminology).
<p>
It's clear that the original matrix entry E is the only nonzero one in
its row and column. But also, for each of the entries in the smaller matrix,
if we follow their row and column to where they meet E's we find zeroes
(since we don't find E). So the result is a monomial matrix.

<p>
4. If M is a monomial matrix, and T any square matrix of the same size, 
show that det MT = det M det T.
<p>

A. To split up the problem, let's note first that every monomial
matrix is the product of a diagonal matrix D and a "permutation matrix" P,
which is a monomial matrix with exactly one 1 in each row and column.
Even better, we can write P as a product of permutation matrices that
just switch two columns.
<p>
For D the statement easily follows from multilinearity. For P a permutation
that just switches two rows it follows by the "switching two columns negates"
theorem we proved in class. Then 
<center>
det MT = det (D PQR T) = det D det PQRT = det D det P det QRT
= det DP det Q det RT = det DPQ det R det T = det DPQR det T = det M det T
</center>
if M is D times some column-switching matrices P,Q,R.

<p>
5. Use questions #2, #3, #4 to show that det AB = det A det B, 
for any two square matrices of the same size.
<p>
A. #3 says that we can write A = PMQ, where P,Q are 
products of positive root matrices, and M is monomial. Then #2,#4 say
<p>
<center> 
det AB = det PMQB = det P det MQB = det P det M det QB = det PM det Q det B
= det PMQ det B = det A det B.
</center>