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

Algebra ->  Matrices-and-determiminant -> 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       Log On


   

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(52814) About Me  (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

             %28A-B%5E%28-1%29%29  is non-degenerate;

             %28A-B%5E%28-1%29%29%5E%28-1%29 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  B%5E%28-1%29.


    Multiply the matrix A%2AB-I by the matrix B%5E%28-1%29  on the right. You will get

    %28A%2AB-I%29%2AB%5E%28-1%29 = A+-+B%5E%28-1%29.


    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.