First slide

System of linear equations

Question

The equations $x + y + z = 6, x + 2 y + 3 z = 10, x + 2 y + λ z = μ$ have unique solution if

Moderate

A

$λ = 3, μ = 10$
B

$λ = 3, μ \neq 10$
C

$λ \neq 3$
D

none

Solution

Augmented matrix is
$[\begin{matrix} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & λ & μ \end{matrix}]$
$\begin{matrix} R_{2} - R_{1} \\ R_{3} - R_{1} \\ ~ \end{matrix} [\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 1 & λ - 1 & μ - 6 \end{matrix}]$
$\begin{matrix} R_{3} - R_{2} \\ ~ \end{matrix} [\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & λ - 3 & μ - 10 \end{matrix}]$
$∴ Given equations have unique solution \Rightarrow λ - 3 \neq 0 \Rightarrow λ \neq 3$

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