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.)
2. Call a square matrix U a positive root matrix 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.
3. Define a monomial matrix 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.
4. If M is a monomial matrix, and T any square matrix of the same size, show that det MT = det M det T.
5. Use questions #2, #3, #4 to show that det AB = det A det B, for any two square matrices of the same size.