# Is there a determinant rule for adding rows of ANOTHER matrix?

I'm aware of the following three rules; each for an elementary operation: *Note || denotes determinant; I find it kind of confusing with abs but that's what the book says.

E1: if the matrix B is A but with rows swapped, $$|B|=-|A|$$

E2: if the matrix B is A but a row is a multiple $$k$$ of another, $$|B|=k|A|$$

E3: if the matrix B is A but a multiple of another row is added/subtracted to another, $$|B| = |A|$$

However, I came across this question to find the determinant of $$C$$ given $$det A = 3$$ and $$det B = -2$$: $$A = \begin{bmatrix} a&b\\ c&d \end{bmatrix}$$ $$B = \begin{bmatrix} e&f\\ c&d \end{bmatrix}$$ $$C = \begin{bmatrix} 2a-e&2b-f\\ 3c&3d \end{bmatrix}$$

I was able to solve it by the "usual way" of just multiplying $$(2a-e)(3d)-(3c)(2b-f)$$ because it's a simple 2 by 2, but I was wondering if there would be a way to use the logic of the elementary ops to "remove row 1 of matrix B from the matrix C" such that then matrix C has rows that are just scalar multiples of A, so then I can easily apply the E2 rule.

• This is the multilinearity of the determinant as a function of its rows. – Lord Shark the Unknown Dec 5 '19 at 3:53
• I have a feeling that's something I haven't covered; I'm going to search it up but could you give a very brief description? (I'm in the first/intro linear algebra class of undergrad) – Five9 Dec 5 '19 at 3:59
• I think you do need to do it in the usual way and then maybe see if you can write this determinant equation using those for $A$ and $B$. Although $C=MA+NB$ for some matrices $M,N$ there is usually no way of 'splitting' this sum. So it is a good question to think about – AnyAD Dec 5 '19 at 4:08
• The appearance of $-2,3$ for determinants and then when row operations are performed is likely what makes the answer bellow to work. So the answer to your question is 'not in general' – AnyAD Dec 5 '19 at 4:14

Now write the determinant as $$\det{C}=6(ad-bc)+3(cf-ed)=12$$ (from your expression for the determinant of $$C$$).