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 k is

Moderate

A

an empty set
B

equal to R
C

equal to 0
D

equal to $R - {0}$

Solution

Augmented matrix is $[\begin{matrix} 1 & 1 & 1 & 2 \\ 2 & 1 & - 1 & 3 \\ 3 & 2 & K & 4 \end{matrix}]$ $~_{R_{3} - 3 R_{1}}^{R_{2} - 2 R_{1,}} [\begin{matrix} 1 & 1 & 1 & 2 \\ 0 & - 1 & - 3 & - 1 \\ 0 & - 1 & k - 3 & - 2 \end{matrix}]$
$\overset{R_{3} - R_{2}}{~} [\begin{matrix} 1 & 1 & 1 & 2 \\ 0 & - 1 & - 3 & - 1 \\ 0 & 0 & k & - 1 \end{matrix}]$
System of equations has a unique solution
$\Rightarrow k \neq 0,$ solution set $= R - {0}$

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