[Math] Gaussian Elimination with unknowns

gaussian eliminationmatrices

I need to use Gaussian Elimination to find values of k so that the following has, (i) no solutions, (ii) infinite solutions (iii) a unique solution
$$A= \left[
\begin{array}{ccc|c}
2&2&0&2\\
0&k&1&1\\
1&2&k&2
\end{array}
\right] $$

I can reduce it down to
$$ \left[
\begin{array}{ccc|c}
1&1&0&1\\
0&k&1&1\\
0&1&k&1
\end{array}
\right] $$
and I understand it has no solutions when its inconsistent. But that isn't the case?
I used the NS(A) and found solutions at k=1 and k=-1 but both of those have NS(A) not equal to 0. So they should be my infinite solutions. But I still can't work out how to find the k values that have no solutions.

Best Answer

You have not fully row-reduced the matrix: you can, for example, swap the last two rows, then subtract $k$ times the second row from the third:

$$\left[\begin{array}{ccc|c}1 & 1 & 0 & 1\\0 & 1 & k & 1\\0 & 0 & 1-k^2 & 1-k\end{array}\right].$$

Now we have three cases:

  • if $1-k^2\neq 0$ the matrix is nonsingular and there is a unique solution to the linear system.
  • if $k=1$, the matrix has rank 2, but the right-hand side is in the column space of the matrix. Therefore there are infinitely many (a one-dimensional family of) solutions.
  • if $k=-1$, the matrix has rank 2 but the right-hand side is not in the column space of the matrix. There are therefore no solutions.
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