Question 1182268
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Your writing in your post is INCORRECT,


THEREFORE,  I edited it in couple of lines below.



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

             {{{(A-B^(-1))}}}  is non-degenerate;

             {{{(A-B^(-1))^(-1)}}} is non-degenerate. Find the inverse of this matrix 
</pre>


This problem was posted to the forum several years ago, &nbsp;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.


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



<pre>
    Since the matrix B is non-degenerated, it has the inverse matrix  {{{B^(-1)}}}.


    Multiply the matrix {{{A*B-I}}} by the matrix {{{B^(-1)}}}  on the right. You will get

    {{{(A*B-I)*B^(-1)}}} = {{{A - B^(-1)}}}.


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


    QED
</pre>


Solved.