Suppose it true for n=k, now consider any (k+1)x(k+1) matrix A, and expand

about the first row:

det(cA) = sum_{r=1 to k+1} (-1)^(r-1) c A_{1,r} det (cB)_{1,r}

where (cB)_{1,r} is the matric obtained from cA by deleting the 1st row and

the r-th col. Then by assumption:

det(cA) = sum_{r=1 to k+1} (-1)^(r-1) c^{k+1} A_{1,r} det (B)_{1,r}

...........= c^{k+1} det(A)

So if its true for n=k its true for n=k+1. It is trivialy true for n=1.

Hence we conclude by mathematical induction that it is true for all

positive integers.

RonL