Algebraic/Matrix Manipulation (linear algebra)

[boings]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Also attached, in blue.


Thanks a lot for any help at all!

 

[mfb]

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?

 

[boings]

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.

The equation Ax= 0 always has the "trivial" solution, x= 0. It has other solutions if and only if its determinant is 0.

 

[boings]

You're right, I should know that basic property :) I've been cramming too much this semester so I tend to forget.

Thanks a lot it makes good sense.