If you use row-reduction to make a n x n matrix A into a triangular matrix then you can take the product of the diagonal to achieve det(A)

and if you multiply a row/column by a non-zero scalar $k$ the effect is $k\det(A)$

also $\det(kA) = k^n\det(A)$

thus therein lies my question.

if I use row reduction then multiply some row by $k$ , call the new triangular matrix B then it becomes $k\det(B)$

so what is the effect on $\det(A)$ because $k\det(B) \neq \det(A)$

is it $k\det(B) = \det(kA) = k^n\det(A)$?