Question

Physically transpose of a matrix can be thought of as the mirror image of the original matrix. Similarly what is the physical meaning of the inverse of a matrix?

Answer 1

Well, in real numbers, the inverse of any real number a was the number a^-1, such that a times a^-1 equaled 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero.

The inverse of a square matrix A, denoted by A^-1, is the matrix so that the product of A and A^-1 is the Identity matrix. The identity matrix that results will be the same size as the matrix A. Wow, there's a lot of similarities there between real numbers and matrices. That's good, right - you don't want it to be something completely different.