For what value of $k$ does system of Equations have infinitely many solutions? Gaussian Elimination - is my approach correct?

Question

$$ x + ky + z = 1$$ $$ - y + z = 2$$ $$x + y + 2z = 3$$

Using Gaussian elimination I reduced the augmented matrix to:

[\begin{bmatrix} 1&1&2&3\\ 0&-1&1&2\\ 0&0&k-2&2k-4\\ \end{bmatrix}]

And so for k = 2 there are infinitely many solutions?

Answer 1

The approach of using Gaussian elimination is for sure correct, but your equation system isn't the one represented by your reduced matrix, because:

1. The equation system has no solution for $k=1$, otherwise it has one unique solution.
2. The system represented by your reduced matrix has infinitely many solutions if $k=2$.