SOLUTION: Suppose A,B and AB _1 are non-degerate and have same order prove that (A-B¹) is non-degerate; (A-B¹)-¹ is non-degenerate find the inverse of this matrix

Question 1182268: Suppose A,B and AB _1 are non-degerate and have same order prove that
(A-B¹) is non-degerate;
(A-B¹)-¹ is non-degenerate find the inverse of this matrix
Answer by ikleyn(52817) (Show Source): You can put this solution on YOUR website!
.

Your writing in your post is INCORRECT,

THEREFORE, I edited it in couple of lines below.

          Suppose A,B and AB-I are non-degenerate and have same order. Prove that

               is non-degenerate;

              is non-degenerate. Find the inverse of this matrix

This problem was posted to the forum several years ago, and I solved it under this link

https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Matrices-and-determiminant.faq.question.1099395.html

Below I copy-pasted that solution here again for your convenience.

..............................

    Since the matrix B is non-degenerated, it has the inverse matrix  .


    Multiply the matrix  by the matrix   on the right. You will get

     = .


    Since both matrices on the left side of the equation are non-degenerated, their product on the right side is non-degenerated, too.


    QED

Solved.

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