How do I prove - If matrix A is similar to matrix B and matrix C is similar to matrix D, then matrix A*C is similar to matrix B*D?

Question

I am generally looking for solving two pairs of n*n matrices. Also, if the nth row of matrix A is in the mth row of matrix C then the nth row of matrix B will also be in the mth row of matrix D. I am providing an example just for reference -

Answer 1

The statement of your title is not even true for the very well behaved class of invertible diagonal matrices. Consider $A=\binom{1~~0}{0~~2}$ and $C=\binom{2~~0}{0~~1}$; these are clearly similar, so if what you say were true one should have that $AC$ and $AA=A^2$ are similar. But $AC=\binom{2~~0}{0~~2}=2I$ which is not similar to any matrix different from it, while $A^2=\binom{1~~0}{0~~4}$.

Answer 2

This is not true. E.g. consider $$ A=D=\pmatrix{1&0\\ 1&0},\ B=C=\pmatrix{0&0\\ 1&1},\ AC=0,\ BD\ne0. $$ This counterexample actually shows more, namely, even if $A$ is similar to $B$, $AB$ can be dissimilar to $BA$.