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 ≠ 0
(b) −1 < k < 1
(c) −2 < k < 2
(d) k = 0
Solution:
(a) $k \neq 0$
For a unique solution, $|A| \neq 0$.
The given system of equations can be written in matrix form as follows:
$\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & k\end{array}\right] \neq 0$
$\Rightarrow 1(k+2)-1(2 k+3)+1(4-3) \neq 0$
$\Rightarrow k+2-2 k-3+1 \neq 0$
$\Rightarrow k \neq 0$
So, the given system of equations has a unique solution if $k$ is not equal to 0 .