I want to find the minimum to the following problem:

$$
min_{X} ||f(X)||_F \tag{1}
$$
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 (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.