a) Suppose that B is an n ¡Á n matrix, and k is a scalar. Give a formula for det (kB) in terms of det B .
b) Show that your formula from (a) is true for all n and for any k.
det (kB) = k^n det B
This is because the determinant is a multilinear function of it's columns (or rows) considered as vectors to the reals. If you multiply one particular colomn of the matrix by k, then the entire determinat get's multiplied by k.
So, if you muliply the whole matrix by k, then that amounts to multiplying each column by k, and thus the determinant picks up a factor k^n.