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

The system of linear equations will have a unique solution if,
$\begin{array}{l} |\begin{array}{r} 1 & 1 & 1 \\ 2 & 1 & - 1 \\ 3 & 2 & k \end{array}| \neq 0 \\ \Rightarrow 1 (k + 2) - 1 (2 k + 3) + 1 (4 - 3) \neq 0 \\ \Rightarrow - k + 2 - 2 \neq 0 \\ \Rightarrow k \neq 0 . \end{array}$

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