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 $k\ne 0$

For unique solution, the determinant formed by co-efficient of $x,yandz$ has to be non zero
$\left|\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right|\ne 0$
$R_1-R_1-R_2,R_2-R_2-R_3$
$\left|\begin{array}{ccc}0& 0& 1\\ 1& 2& -1\\ 1& 2-k& k\end{array}\right|\ne 0$
$\Rightarrow (2-k)-2\ne 0$
$-k\ne 0$
$k\ne 0$