I want to find the minimum to the following problem:

$$ \min_{X} \|f(X)\|_F \label{1}\tag{1} $$ where $X$ is a rectangular matrix, and $f$ is a function of it, involving other matrices, the norm is Frobenius norm.

I want to know if the solution of problem \eqref{1} is equivalent to the following problem: $$ \min_{X} \|f(X) A\|_F \tag{2} $$ where $A$ is a rectangular matrix.

Are the two problems equivalent? if not, are there some properties on $A$ so that they become equivalent?

Thank you in advance.