Gaussian Elimination with unknowns

Question

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.

Answer 1

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.