I had a true or false quiz in a linear algebra course, one of the statements read
the rank of a matrix and its inverse are always equal
I answered true but the professor said it is false, he said that not all matrices have an inverse, but, I tough that since the statement says "and its inverse" it had an inverse since the statement say it; so my question would be, is the statement fine by being false or is it an error in the thinking of the solution of my professor, thanks
This is a terrible question as it's more a question about semantics than about mathematics.
There are several reasonable ways of interpreting the question:
A) "Given a matrix $M$, does there exist an inverse of $M$ with rank equal to that of $M$"?
The answer to this question is trivially false; for some matrices the inverse does not exist.
B) "Given a matrix $M$ and its inverse $M^{-1}$, do $M$ and $M^{-1}$ have the same rank?"
This is true, equally trivially: if $M$ has an inverse then it is square and full rank, and $M^{-1}$ likewise.
C) "Given a matrix $M$, does its Moore-Penrose pseudoinverse $M^{+ }$ have the same rank?"
Now this is an interesting question; it is worth thinking about why the answer is also "true" (one way to approach it is to look at the SVD of $M$).