Quantifying the "usefulness" of Penrose-pseudoinverse

Question

1. Given a real MxN matrix $A$ (non-invertible), is there a way to quantify the degree of "usefulness" of the Penrose-pseudo inverse $A^+$? Is the distance $||AA^+ - I||^2$ a useful measure? i.e., can you quantify the "degree of invertability" of a non-invertible $A$?
2. Given such a measure, is the problem of finding the "worst" real MxN matrix (most non-invertible) under the constraint of $A$ having a rank $r$ solvable?

Thanks!

Answer 1

EDIT: $A A^+$ is always an orthogonal projection, so $\|A A^+-I\|$ is always $0$ or $1$.