When calculating determinants it can be nice to multiply a row by a number or to add one row to another (your basic row operations). Each has an easy to understand effect on the determinant. Today I ran across a type of row operation whose effect on the determinant is a lot less clear to me.
Take an $n \times n$ matrix $M$ and an $n \times n$ matrix $A$. While calculating the determinant of $M$, I might mess around with the rows of $M$. Every time I mess with the first row of $M$, I would like to be able to change my mind and mess with the second row of $M$ instead.
If I multiply the first row of $M$ by $2$ to get M', or if I multiply the second row of $M$ by $2$ to get $M″$, I get the same determinants, \det(M') = \det(M″). This does not depend on the matrix $M$; I always have that $2$ times the first row is equivalent to $2$ times the second row.
For instance, if $M = \left(\begin{smallmatrix}a& b \\ c& d \end{smallmatrix}\right)$ then M' = \left(\begin{smallmatrix}2a& 2b \\ c& d \end{smallmatrix}\right) and M'' = \left(\begin{smallmatrix}a& b \\ 2c& 2d \end{smallmatrix}\right) and \det(M') = \det(M″) = 2\det(M) = 2ad−2bc.
Can I do something similar with "$A$" instead of "$2$"?
If I multiply the first row of $M$ by $A$ to get M', then what matrix $B$ do I need to multiply the second row of $M$ by to get M'' so that \det(M')=\det(M'') have the same determinant?
I would like the answer to be depend only on $A$, not on $M$. I calculated an answer for 2×2 matrices, but I'm not happy with it.
For instance, if $A = \left(\begin{smallmatrix}2& 0 \\ 0& 1 \end{smallmatrix}\right)$, and $M = \left(\begin{smallmatrix}a& b \\ c& d \end{smallmatrix}\right)$, then M' = \left(\begin{smallmatrix}2a& b \\ c& d \end{smallmatrix}\right) and \det(M') = 2ad−bc. I can choose $B = \left(\begin{smallmatrix}1& 0 \\ 0& 2 \end{smallmatrix}\right)$ so that M'' = \left(\begin{smallmatrix}a& b \\ c& 2d \end{smallmatrix}\right) and \det(M'') = 2ad−bc = \det(M').
However, I'd like to understand more clearly the relationship between $A$ and $B$. Perhaps there is a fair amount of freedom in choosing $B$ and I have chosen a bad one in my work. What is a natural choice of $B$ (for general $A$)?