Working on this problem. Actually 2 problems, involving Gaussian elimination. Both of them say

Find the values of $\displaystyle k$ such that the system has (a) a unique solution (b) no solution and (c) infinitely many solutions

I am required to use Gaussian Elimination

The first problem is

$\displaystyle x+ky = -1$

$\displaystyle kx+y = 1$

My solution is as follows, but not sure if correct (not sure how to do latex for this). I have seperated each entry by a comma (,)

row 1 = 1, k, -1

row 2 = k, 1, 1

-k * R1 + R2 = 0, -(k^{2})+1, k+1

(a) if k = -1 R3 = 0, 0 and has infinitely many solutions

(b) if k = 1 R3 = 0, 2 and is inconsistant

(c) if k != 1 and k != -1 the system has a unique solution

I'm not sure if that is correct. I've gone through the problem several times, and come up with the same outcome everytime, but I'm new to this. I did really well at basic Matrix stuff, but this is the next paper after the intro one.

The second problem is

$\displaystyle x-2y+3z = 2$

$\displaystyle x+y+z = k$

$\displaystyle 2x - y + 4z = k^2$

I've gone through this a whole lot of times (I need to use Gaussian elimination) and end up with some unfactorable equation at the end. Where should I start with this problem? I don't want the answer, just some help.

What I have been doing is laying it out in a compact matrix and then multiplying the rows to eliminate the x and y values.

Here is my working - not sure if it is correct.