Question

For what values of $k$, the system of linear equations
$x+y+z=22x+y-z=33x+2y+kz=4$
have a unique solution?

Answer 1

First we arrange the system of linear equation in matrix form, i.e.,

$AX=B$

Where $A=\left[\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right];X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right];B=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]$

So for a unique solution $|A|\ne 0$

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

$\Rightarrow 1(k+2)-1(2k+3)+1(4-3)\ne 0$

$\Rightarrow k+2-2k-3+1\ne 0$

$\Rightarrow -k\ne 0$

$\Rightarrow k\ne 0$

so for any value of $k\ne 0$ the given system of linear equation has a unique solution.