Question

For what values of k, the system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = 4 has a unique solution ?

Answer 1

Let A = $\left(\begin{array}{c}111\\ 21-1\\ 32k\end{array}\right)$, which is matrix formed by the coefficients of x, y & z in the system of eqs.

For unique solution, $|A|\ne 0so,\left|\begin{array}{c}111\\ 21-1\\ 32k\end{array}\right|\ne 0$

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