Determine the value(s) of k for which the system
-2 1 -1 x 1
4 2 k . y = -4
k -1 1 z 2
(please excuse my formatting)
The Attempt at a Solution
On my first attempt I tried making an augmented matrix and row-reducing it, but it started to get really hairy and taking too long, (this is an exam question), so I figured there must be an easier
way of solving it.
One thing I noticed is that if I set k=2 then the first and third rows of the matrix are multiples of each other (r1=-r3), however the corresponding values in the result vector aren't, this would lead me to suspect that k=2 gives me an inconsistent system.
IF I set k=-2 I notice the 2nd and 3rd rows are scalar multiples of each other (including the result). So I guess this is a case of linear dependence somewhere. In other words if I row-reduced the augmented matrix I'd have a row of zeros and case of one of the x/y/z being a multiple of another.
Then I guess that leaves all other values of k to mean 1 solution
Am I approaching this correctly? Have I just solved this? I have a test on Thursday and I'd appreciate any help or tips for looking at matrices and finding clues and short-cuts to solving them.
If there's any experts on Linear Algebra reading, how would you approach this question? I'd love to know