Answer
(a) there three two equations in two variables.
(b) the system is consistent if and only if $k+6= 0$, that is $k=-6$.
If the matrix is the coefficient matrix of homogeneous system, we have
(a) there are three equations in three variables.
(b) the system is consistent for any value of $k$. It has at least the trivial solution.
Work Step by Step
The augmented matrix is given by
$$ \left[ \begin {array}{ccc} 2&-1&3\\ -4&2&k
\\ 4&-2&6\end {array} \right].
$$
Multiply the first row by $2$ and adding it to the second row, adding the second row to the third row, we get
$$\left[ \begin {array}{ccc} 2&-1&3\\ 0&0&k+6
\\ 0&0&k+6\end {array} \right].
$$
Adding $-1$ times the second row to the third row we get
$$\left[ \begin {array}{ccc} 2&-1&3\\ 0&0&k+6
\\ 0&0&0\end {array} \right].
$$
Now, we have
(a) there three two equations in two variables.
(b) the system is consistent if and only if $k+6= 0$, that is $k=-6$.
If the matrix is the coefficient matrix of homogeneous system, we have
(a) there are three equations in three variables.
(b) the system is consistent for any value of $k$. It has at least the trivial solution.
