Multiplying with $D_a$ from left makes one singe elementary row operation on $M$. More specific $D_aM$ is different from $M$ just in the first row, and if I denote the first row with index 1, then if $M$ has the first row $M_1$ then $D_aM$ has $aM_1$. This property comes from the construct of matrix multiplication. You read more about elementary row operations here and here.
In general case elementary row operations will not preserve eigenvalues. There are special situations, for example if you get a similar matrix $B$ with row operations from $A$, so you get $A \sim B$. In this case the eigenvalues are the same, but eigenvectors not. But in general case row operations may completely change eigenvalues. You can find examples for this here. Because for small sizes we get the eigenvalues from determinants in general, you can find results in this topic here. Also this article could be interesting, but it is a little bit far from here.
In your special case you can get results only if $M$ has a propery that you can handle. For example if $M$ is a triangular matrix then one of its $\lambda$ eigenvalue changes to $a \lambda$. There are other matrix classes which you can treat by algebraic tricks, but in general I think you can say nothing about this mathematics.