# Algebraic/Matrix Manipulation (linear algebra)

1. Jan 5, 2013

### boings

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

Also attached, in blue.

Thanks a lot for any help at all!

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2. Jan 5, 2013

### Staff: 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?

3. Jan 6, 2013

### 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?

4. Jan 6, 2013

### 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.

5. Jan 6, 2013

### 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.