Transform $⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 2 2 & 1 & 3 3 & 2 & 4 \end{matrix} ⎤ ⎥ ⎦$ into an upper triangular matrix by suitable column transformations

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Solution

$\begin{matrix} c_{1} \to c_{1} - \frac{2}{3} c_{3} c_{2} \to c_{2} - \frac{1}{3} c_{3} \end{matrix}$ $⎡ ⎢ ⎣ \begin{matrix} - v_{3} & - 5 / 3 & 2 0 & 0 & 3 1 / 3 & 2 / 3 & 4 \end{matrix} ⎤ ⎥ ⎦$ ; $\begin{matrix} c_{1} \to 3 c_{1} c_{2} \to 3 c_{2} \end{matrix}$ $⎡ ⎢ ⎣ \begin{matrix} - 1 & - 5 & 2 0 & 0 & 3 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎦$
$c_{1} \to c_{1} - \frac{1}{2} c_{2} ⎡ ⎢ ⎣ \begin{matrix} 3 / 2 & - 5 & 2 0 & 0 & 3 0 & 2 & 4 \end{matrix} ⎤ ⎥ ⎦ c_{2} \to c_{2} - \frac{1}{2} c_{3} ⎡ ⎢ ⎣ \begin{matrix} 3 / 2 & - 6 & 2 0 & - 3 / 2 & 3 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow$ An upper triangular matrix.

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