Question

Let $S$ be the set of all real values of $k$ for which the system of linear equations
$x+y+z=22x+y-z=33x+2y+kz=4$
has a unique solution. Then $S$ is

• A. an empty set
• B. equal to $\left\{0\right\}$
• C. equal to $R$
• D. equal to $R-\left\{0\right\}$

Answer 1

The correct option is D equal to $R-\left\{0\right\}$

For unique solution,
$\left|\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right|\ne 0\Rightarrow k+2-2k-3+1\ne 0\Rightarrow k\ne 0$