Suppose the vectors x _ 1 , x _ 2 and x _ 3 are the solutions of...

Let A = \left[\begin{cases}{\alpha}_{1} & {\alpha}_{2} & {\alpha}_{3} \\ {\beta}_{1} & {\beta}_{2} & {\beta}_{3} \\ {\gamma}_{1} & {\gamma}_{2} & {\gamma}_{3}\end{cases}\right]. Now satisfying ${x}_{1}, {x}_{2}, {x}_{3}$ in $Ax = b$
A{x}_{1} = {b}_{1} ⇒ \left[\begin{cases}{\alpha}_{1} & {\alpha}_{2} & {\alpha}_{3} \\ {\beta}_{1} & {\beta}_{2} & {\beta}_{3} \\ {\gamma}_{1} & {\gamma}_{2} & {\gamma}_{3}\end{cases}\right]\left[$\begin{array}{ll}1 \\ 1 \\ 1\end{array}$\right] = \left[$\begin{array}{ll}1 \\ 0 \\ 0\end{array}$\right]

⇒ \left\{$\begin{array}{ll}{\alpha}_{1} + {\alpha}_{2} + {\alpha}_{3} = 1 \\ {\beta}_{1} + {\beta}_{2} + {\beta}_{3} = 0 \\ {\gamma}_{1} + {\gamma}_{2} + {\gamma}_{3} = 0\end{array}$\right \ldots \left(i\right)

Now

A{x}_{2} = {b}_{2} ⇒ \left[\begin{cases}{\alpha}_{1} & {\alpha}_{2} & {\alpha}_{3} \\ {\beta}_{1} & {\beta}_{2} & {\beta}_{3} \\ {\gamma}_{1} & {\gamma}_{2} & {\gamma}_{3}\end{cases}\right]\left[$\begin{array}{ll}0 \\ 2 \\ 1\end{array}$\right] = \left[$\begin{array}{ll}0 \\ 2 \\ 0\end{array}$\right]

⇒ \left\{$\begin{array}{ll}2{\alpha}_{2} + {\alpha}_{3} = 0 \\ 2{\beta}_{2} + {\beta}_{3} = 2 \\ 2{\gamma}_{2} + {\gamma}_{3} = 0\end{array}$\right \ldots \left(ii\right)

And satisfying ${x}_{3}$ gives,

${\alpha}_{3} = 0$, ${\beta}_{3} = 0 and${\gamma}_{3} = 2$</p><p>Now using \left(i\right)\text{and}\left(ii\right)$

${\alpha}_{2} = 0, {\beta}_{2} = 1, {\gamma}_{2} = - 1$ and ${\alpha}_{1} = 1, {\beta}_{1} = - 1, {\gamma}_{1} = - 1$
Thus, A = \left[\begin{cases}1 & 0 & 0 \\ - 1 & 1 & 0 \\ - 1 & - 1 & 2\end{cases}\right] ⇒ \left|A\right| = 2