SOLUTION: Please help Suppose A, B and AB-I are non-degenerate and have same order. Prove that (1) {{{A - B^-1}}} is non-degenerate. (2) {{{(A - B^-1)^-1 - A^-1}}} is non-degenerate.

Algebra ->  Matrices-and-determiminant -> SOLUTION: Please help Suppose A, B and AB-I are non-degenerate and have same order. Prove that (1) {{{A - B^-1}}} is non-degenerate. (2) {{{(A - B^-1)^-1 - A^-1}}} is non-degenerate.       Log On


   

Question 1099395: Please help
Suppose A, B and AB-I are non-degenerate and have same order. Prove that
(1) A+-+B%5E-1 is non-degenerate.
(2) %28A+-+B%5E-1%29%5E-1+-+A%5E-1 is non-degenerate. find the inverse of this matrix.
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
1)  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