Question

The system of linear equations $x+y+z=2,2x+y-z=3,3x+2y+kz=4$ has a unique solution if

• A. $k\ne 0$
• B. $-1<k<1$
• C. $-2<k<2$
• D. $k=0$

Answer 1

The correct option is A $k\ne 0$

The liner system will have unique equation when

$\left|\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right|\ne 0$

Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$

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

$\Rightarrow -(k-3)-3\ne 0\Rightarrow -k\ne 0$