Given an n x n matrix A, what happens to its determinant when each of its columns are multiplied by its column number?
I am teaching myself linear algebra with Elementary Linear Algebra by Anton and Rorres. I don't understand the new set of operations that are permissible for solving determinants. This problem in particular stumped me.
If a row or column of a n x n matrix $ A$ is multiplied by k, then the determinant of the new matrix $B$ is
$$ det(B)=k \text{ } det(A) $$
The proof of the above property comes from the definition of determinant. You can look it up Change in determinant when multiplying row of a matrix
By this property, if each of the columns are multiplied by its column number, then
$$ det(B)= (1 * 2 * 3 *........* n) det(A) $$
