Prove that det(kA)= k det(A) for any A M_nxn(F).
Is this proof correct?
det(AB) = det(A)det(B) so
det(kA) = det(kI[n]A) = det(kI[n])det(A) = k^n det(I[n]) det(A) = k^n (I) det(A) = k^n det(A)
You could alternatively note that if that and thus
Remark: Really the operative thing here is that the determinant of a matrix over can really be thought of as a -linear form
Where we've interpreted
and thus if we see that the interpretation of your question is really to show that but this is apparent by the -linearity.