Determinates. Expansion by minors, special case for n = 1, 2, special
case for lots of zeros, special case using one kind of row operation 
Ri += cRj which does not change the det. (The other two operations change
the value of the determinate.  Do not use special 3x3 method.

Given an n x n Matrix A and an entry aij in row i and column j, the
minor of aij is the n-1 x n-1 matrix Aij formed by deleting row i and
column j from A.

For any column j: det A = sum from i=1..n (-1)^(i+j) aij det Aij
For any row i: det A = sum from j=1..n (-1)^(i+j) aij det Aij

The signs form a easy to remember matrix
[ + - + - ...
[ - + - + ...
[ + - + - ...
[ .
[ .
[ .

The final thm's some definitions come later, but invertible/non-invertable
and singular/non-singular were done today.

Thm. If A is an n x n matric, then the following are equivalent
1. The matrix equation A X = B has a unique solution
2. The matrix A has an inverse A^(-1) [so that A A^(-1) = A^(-) A = I].
3. The matrix A is invertible
4. rref(A) = I
5. The columns of A are linearly independent.
6. The rows of A are linearly independent
7. rref(A) has as many pivots as unknowns
8. The matrix A is non-singular
9. The determinate of A, det(A) not= 0
10. The rank of A is n
11. Zero is not an eigenvalue of A

Thm. If A is an n x n matric, then the following are equivalent
1. The matrix equation A X = B has 0 or infinitely many  solutions
2. The matrix A does not have an inverse A^(-1) 
3. The matrix A is non-invertible
4. rref(A) not = I
5. The columns of A are linearly dependent.
6. The rows of A are linearly dependent
7. rref(A) has as fewer pivots than unknowns
8. The matrix A is singular
9. The determinate of A, det(A) = 0
10. The rank of A is k < n
11. Zero is an eigenvalue of A


Scilab commands
	xarrow (arrowdemo)
	plotquad
	fixing the location scilab looks for exec files. 
		I downloaded plotquad.in to the desktop but scilab wasn't looking there
		I typed to pwd to see that scilab was looking at T:\
		I typed cd Desktop to move to T:\Desktop (where the file was)