Question

Show that matrix 8243 is non singular

Answer 1

Answer:

Let n be a positive integer and let F be a field. Let A∈Mn×n(F) be a matrix for which there exists a matrix B∈Mn×n(F) satisfying I+A+AB=O. Show that A is nonsingular.

Since I+A+AB=O, we can get

I+A(I+B)=0

A[−(I+B)]=I

I know it seems −(I+B) is the inverse of A, however, I am not sure how to get −(I+B)A=I. I may choose a wrong way to solve this problem. I am really stuck with this question