# What is the physical significance of inverse of a matrix? [closed]

I was asked this question in an interview. Though I tried my best to answer the question in whatever way I could (I was explaining in terms of mathematics), the professor looked upset.

Any idea?

The professor was not interested in mathematics/equation/properties.

## closed as too broad by whuber♦Sep 13 '16 at 15:39

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• What is the "physical significance" of the matrix in the first place? Unless you provide such a context, there are too many possible answers, because matrices are used to represent a huge number of possible objects and phenomena. – whuber Sep 13 '16 at 15:39
• What is the physical significance of 2? ... Sometimes (or rather often) interviews are just bad. People think of random questions and insist there is only one random answer. If it was a company, it'd be glad to not work for such a person. – Gerenuk Sep 13 '16 at 18:35

## 1 Answer

Matrix Inverse in Terms of Geometry: If a matrix works on a set of vectors by rotating and scaling the vectors, then the matrix's inverse will undo the rotations and scalings and return the original vectors.

If the first linear transformation is not unique, there are several ways to do the transformation and you cannot determine that path you need to take to reverse the transformation. In terms of geometry that means that the vectors you're scaling/rotating are in some sense so alike that you can reproduce a specific result by combining the vectors in more than one way. I believe in terms of statistics we'd refer to that as multicollinearity. If the transformation is not unique then you have a singular matrix, and you need to apply specific rules governing how you interpret the transformation in order to generate the inverse.