I am having some trouble thinking about properties of the determinant.
I understand why it is true that if $B$ is the matrix obtained from an $n \times n$ matrix $A$ by multiplying a row by a scalar $k$ then $$\det(B)=k\det(A),$$ but I don't understand why if you multiply every row then you get $$\det(B)=k^{n}\det(A).$$
I am confused because I thought you only do a row expansion along one row or column anyways to calculate the determinant? I mean, how does the expansion take account for this, what is the intuition, etc?
Thank you in advance.
Best Answer
There are a few ways to see this. If you scale each row by $k$ one at a time, you'll pick up a factor of $k$ each time from the formula you have above ($\det B = k\det A$). Geometrically, the determinant is a way to measure volume where the rows (or columns) correspond to the sides of a parallelepiped. If you take a unit cube and scale the whole thing by $k$, then you now have a $k\times\cdots\times k$ cube. The volume of this cube is now $k^n$, whereas your original had volume $1$. A similar argument works for a general parallelepiped.
If you want to appeal to the cofactor expansion, then each cofactor matrix has picked up a factor of $k$. If you continue to break the cofactors down accordingly, you'll end up picking up a factor of $k^n$ multiplying the original determinant. (I'm hiding an induction argument here, but the general idea is there.)