Get Expert Help with Invertible Matrices: Tips and Techniques

[FancyChancey]

Hi all. I'm having a real tough time with this question. I don't know where to begin and how to go about the question. Can someone point me in the right direction please?

(Problem is on the attachment)

 

[HallsofIvy]

haven't we seen this before?

a) If A is invertible, then (A+ 2A^-1) is invertible.
When does there exist a matrix, B, such that (A+ 2A^-1)B= I?
If I remember correctly, a matrix is invertible if and only if it's determinant is not 0. What is the determinant of A+ 2A^-1?

b) If a real matrix B satisfies B²- 2B+ 1= 0, then B-I cannot be invertible.
B²- 2B+I= (B- I)²

c) There is a non-invertible matrix, B, satisfying B²- 2B+ I= 0.
As I said, B²-2B+ I= (B-I)². Therefore B= ? or ?. Is either of those non-invertible?