First slide

System of linear equations

Question

If the system of equations $x - k y - z = 0, k x - y - z = 0, x + y - z = 0$ has a non-zero solution, then the possible values of $k$ are

Moderate

A

$- 1, 2$
B

$1, 2$
C

$0, 1$
D

$- 1, 1$

Solution

As the given system has a non-zero solution,
$0 = |\begin{array}{ccc} 1 & - k & - 1 \\ k & - 1 & - 1 \\ 1 & 1 & - 1 \end{array}| = |\begin{array}{ccc} 1 + k & - k - 1 & - 1 \\ 1 + k & - 2 & - 1 \\ 0 & 0 & - 1 \end{array}|$
[Using $C_{1} \to C_{1} - C_{2}, C_{2} \to C_{2} + C_{3}$ ]
$\begin{array}{ll} \Rightarrow 0 = (- 1) [(1 + k) (- 2) - (1 + k) (- k - 1)] \\ \Rightarrow 0 = (1 + k) (- 2 + k + 1) \Rightarrow k = - 1, 1 \end{array}$

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