That is not a trivial problem, requires a large proof based on linear maps. Is it homework?.
Let M be a nxn matrix and A is a mxm and C is kxk square matrix.
where O is made of all entries equal to zero and B is any matrix.
Prove det(M) = det(A) det(C)
the only way i can think of it is to expand about the first column but it seems way too tedious. and we only have learnt the definition of determinant and Cramer's rule.
This is trivial if are triangular since if deonts the diagonal entries
Recall though that is continuous (where is given the topology as a subspace of ) and thus continuous on . That said, I'm fairly positive (but not absolutely positive) that
is dense in .