# Algebraic/Matrix Manipulation (linear algebra)

## Homework Statement

I have attached the relevant question as an image (for sake of ease)

## The Attempt at a Solution

Also attached, in blue.

Thanks a lot for any help at all!

#### Attachments

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mfb
Mentor
You could consider ##B^k Y = 0##. Which property does B need to get non-trivial solutions for Y? How is that related to a?
##Y=(B-I_2)X##. How do you get Y=0 (which is a solution to the equation above) with non-trivial X? How is that related to a?

Perhaps the very beginning is where I'm having trouble, I don't know what times a matrix would be equal to zero. Is it the transpose? Or perhaps a matrix whose determinant is zero?

HallsofIvy
If you have studied matrices at all then you should know this basic property: the equation Ax= y has a unique solution if and only if A is invertible: $x= A^{-1}y$. And that is only true if A has non-zero determinant.