Suppose that we have a $p$ dimensional square matrix $A$ whose rank is less than $p$. We know that such a matrix cannot have an inverse and there are several different ways to prove that the $A$ does not have an inverse.
However, I am struggling to obtain an intuition behind why the inverse does not exist. I considered the following ideas to generate an intuition but failed to do so.
The matrix $A$ can be viewed as a set of transformations such as scaling, translation, rotation etc. Thus, when we apply $A$ to a vector in $p$ dimensions it always maps the $p$ dimensional vector to a vector in a subspace spanned by $A$ if it is less than full rank. Lack of an inverse implies that we cannot reverse the transformations. Why not?
The columns of $A$ span only a subspace of $R^p$ if it is less than full rank. Thus, a transformation such as $A y$ takes a vector from $R^p$ to a vector that always belongs to that subspace. The above viewpoint did not help in obtaining an intuition either.
Is there a way to obtain an intuition as to why a rank deficient matrix does not have an inverse?