Algebraic/Matrix Manipulation (linear algebra)

  • Thread starter boings
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  • #1
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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!
 

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  • #2
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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?
 
  • #3
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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?
 
  • #4
HallsofIvy
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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: [itex]x= A^{-1}y[/itex]. 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.
 
  • #5
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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.
 

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