Find for which values of $k$ the system has none, one or infinitely many solutions

Question

Given the following linear system

$$\left[\begin{array}{cccc|c} k & 0 & 1 & 2 & 0\\ -1 & 1 & k & 1 & 0\\ 1 & k & 1 & 3 & 0\end{array}\right]$$

find for which values of $k$ the system has none, one or infinitely many solutions.

I got to

$$\left[\begin{array}{cccc|c}1&k&1&3&0\\0&1+k&1+k&4&0\\0&-k^2&1-k&2-3k&0\end{array}\right]$$

but now I'm stuck. How do I eliminate $-k^2$ using $1+k$?

This is not the first time I got a question like this, how do I solve it?

Answer 1

To perform the second step in row reduction, you need to find the first non-vanishing element in the second row. That would be $k+1$ iff $k\neq-1$. In that case multiply the second row by $k^2$ and the third one with $k+1$ and add both.

Now if $k=-1$ you get $$\left[\begin{array}{cccc|c}1&-1&1&3&0\\0&0&0&4&0\\0&-1&2&5&0\end{array}\right],$$ which is easily solved.

Answer 2

You’ll have to do a case analysis to proceed with the row-reduction. Dividing the second row by $1+k$ is legal as long as $k\ne1$, which you’ll have to deal with separately. You could instead proceed by swapping the last two rows and dividing the second row by $-k^2$, but that gives you a different special case, $k=0$ to deal with.

However, the highlighted question can be answered without doing any calculations whatsoever. The system is homogeneous, so it has at least one solution: the zero vector. Moreover, the system is underdetermined, so it has an infinite number of solutions for all $k$.

Now, if you were asked for the dimension of the solution space for different values of $k$, that would be a different matter. You’d have to do some calculations to find values of $k$ for which the null space of the coefficient matrix is more than one-dimensional. As an alternative to row-reduction, you could examine the $3\times3$ minors of the coefficient matrix. It turns out that there’s no value of $k$ for which they all vanish, so the matrix has full rank for all $k$.