x-2y-z-8-3x-2y-z-10-4x-3y-2z-4-Solve-with-the-help-of-matrix-

Question Number 204056 by hardmath last updated on 05/Feb/24

{\begin{cases} x + 2 y + z = 8 \\ 3 x + 2 y + z = 10 \\ 4 x + 3 y - 2 z = 4 \end{cases}

Solve with the help of matrix

Answered by AST last updated on 05/Feb/24

[\begin{matrix} 1 & 2 & 1 \\ 3 & 2 & 1 \\ 4 & 3 & - 2 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 8 \\ 10 \\ 4 \end{matrix}] \underset{R_{3} - 4 R_{1}}{\overset{R_{2} - 3 R_{1}}{\Rightarrow}} [\begin{matrix} 1 & 2 & 1 & 8 \\ 0 & - 4 & - 2 & - 14 \\ 0 & - 5 & - 6 & - 28 \end{matrix}]

\underset{4 R_{3} - 5 R_{2}}{\overset{2 R_{1} + R_{2}}{\Rightarrow}} [\begin{matrix} 2 & 0 & 0 & 2 \\ 0 & - 4 & - 2 & - 14 \\ 0 & 0 & - 14 & - 42 \end{matrix}] \overset{- 7 R_{2} + R_{3}}{\Rightarrow} [\begin{matrix} 2 & 0 & 0 & ∣ & 2 \\ 0 & 28 & 0 & ∣ & 56 \\ 0 & 0 & - 14 & ∣ & - 42 \end{matrix}]

\Rightarrow 2 x + 0 y + 0 z = 2 \Rightarrow x = 1; 0 x + 28 y + 0 z = 56 \Rightarrow y = 2

0 x + 0 y - 14 z = - 42 \Rightarrow z = 3 \Rightarrow (x, y, z) = (1, 2, 3) .

Answered by MikeH last updated on 05/Feb/24

(\begin{matrix} 1 & 2 & 1 \\ 3 & 2 & 1 \\ 4 & 3 & - 2 \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} 8 \\ 10 \\ 4 \end{matrix})

(\begin{matrix} x \\ y \\ z \end{matrix}) = {(\begin{matrix} 1 & 2 & 1 \\ 3 & 2 & 1 \\ 4 & 3 & - 2 \end{matrix})}^{- 1} (\begin{matrix} 8 \\ 10 \\ 4 \end{matrix})

\Rightarrow (\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} - \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{5}{7} & - \frac{3}{7} & \frac{1}{7} \\ \frac{1}{14} & \frac{5}{14} & - \frac{2}{7} \end{matrix}) (\begin{matrix} 8 \\ 10 \\ 4 \end{matrix}) = (\begin{matrix} 1 \\ 2 \\ 3 \end{matrix})

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