Is there any difference between hypothesis testing $A\beta = a$ and $\beta = A^{-1}a$? Let $\beta,a_0 \in \Bbb{R}^d$. My professor said we wanted to test 
$$H_0: A\beta = a$$
$$H_1: A\beta \ne a$$
However, I don't understand why she would write it like this and I haven't seen hypothesis tests before where the parameter of interest is multiplied by a matrix. Assuming $A$ is invertible, then we could just test 
$$H_0: \beta = A^{-1} a$$
$$H_0: \beta \ne A^{-1} a$$
So I can only see an advantage of this if $A$ was not invertible. Are there situations where that is useful or is this just a bad way to notate hypothesis testing? 
EDIT: 
Here's the context



 A: If you wish to perform a constrained linear regression with linear equality constraints of the form $A\beta = a$, then tests such as the likelihood ratio test are more naturally formulated in this way.  This is because the constrained OLS estimator $\beta^*$ can be written in terms of the unconstrained estimator $\hat{\beta}$, $A$, and $a$:
$$\beta^* = \hat{\beta} + (X'X)^{-1}A'(A(X'X)^{-1}A')^{-1}(a-A\hat{\beta})$$
$\beta^*$ is best linear unbiased, given the constraints.  Our test is then easily seen to be that $a-A\beta=0$, or, slightly rearranged, that $A\beta = a$, whereas the use of the inverse does not jump so readily to the eye.
Also note that it may be that $A$ is not square, and therefore not invertible, for example, if you are constraining $\hat{\beta}_1 + \hat{\beta}_2 = 0$ in your regression and want to test the corresponding hypothesis about $\beta_1$ and $\beta_2$, $A$ would be $1 \times k$, where $k$ is the length of $\beta$.  This may not apply to your case, judging from the first line of the question, but in the more general case it is definitely an issue.  
A: It is often the case that $A$ is not a square matrix, and so not invertible.  However, assuming that $A$ is actually invertible, the statement $A \beta = a$ is equivalent to $\beta = A^{-1} a$, and so your formulation of the test is legitimate (though unusual).  Regardless of which form you use to state the hypotheses, the actual content of the test is determined by finding an appropriate test statistic and using it to compute the p-value.  This exercise is not affected by which of the two equivalent forms you use to state the hypotheses.
