First slide

System of linear equations

Question

The system of linear equations $x + y + z = 2, 2 x + y - z = 3, 3 x + 2 y + k z = 4$ has a unique solution, If

Easy

A

$k \neq 0$
B

$- 1 < k < 1$
C

$- 2 < k < 2$
D

$k = 0$

Solution

$\begin{array}{l} Δ = |\begin{matrix} 1 & 1 & 1 \\ 2 & 1 & - 1 \\ 3 & 2 & k \end{matrix}| \neq 0 \\ c_{2} \to c_{2} - c_{1}, c_{3} \to c_{3} - c_{1} \\ \Rightarrow |\begin{matrix} 1 & 0 & 0 \\ 2 & - 1 & - 3 \\ 3 & - 1 & k - 3 \end{matrix}| \neq 0 \\ \Rightarrow (k - 3) + 3 \neq 0 \Rightarrow k \neq 0 \end{array}$

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