MATRIX QUESTION Prove that if A is an n n matrix and k is a

MATRIX QUESTION
Prove that if A is an n × n matrix and k is a scalar, then det(kA) = kn detA

Solution

First, let\'s recall what multiplication of a matrix by a scalar means: When we multiply kAkA, where AA is an n×nn×n matrix, and kk a scalar, then every entry aijaij of matrix AA is multiplied by kk: i.e. aijkaijaijkaij for each aijaij. That means for each row ii 0in0in, we can factor out kk.

Now...Recall the elementary row operations you\'ve learned, and how each one of them affects the determinant of the matrix on which it is operating. Specifically, when any one row is multiplied by the scalar kk, the determinant of AA becomes kdetAkdetA. So given that scalar multiplication of a squaren×nn×n matrix is equivalent to \"row operating\" on nn rows (by multiplying each row by the scalar kk), we can conclude that

det(kA)=kkkntimesdetA=kndetA