Linear Algebra: Matrix doubts

Question

In A.A^-1 = A^-1.A

How (A^-1)^-1 = A ?

 

Tell me the process

Answer 1

Another way to look at this:

 

If B^-1 exists, then it is by definition that matrix which satisfies B^-1B = BB^-1 = I

 

Suppose (A^-1)^-1 exists.  Call it C.  Then by definition, C is that matrix which satisfies

 

CA^-1 = A^-1C = I

 

Now ask yourself, what is the only matrix that can satisfy this relation?  Only C = A works, so (A^-1)^-1 = A.

Answer 2

A matrix can have an inverse only if it's "square" (same # rows and columns). IF the inverse of matrix A exists, then the product of A and its inverse is commutative, and equals the identity matrix.

 

Taking the inverse of matrix A, and then inverting that inverse, always gives A.