Solve this following
Question:

Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear equations, $A x=b$ when the vector $b$ on the right side is equal to $b_{1}, b_{2}$ and $b_{3}$ respectively. If

$x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x_{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x_{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b_{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$

$\mathrm{b}_{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]$ and $\mathrm{b}_{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right]$, then the determinant of

$\mathrm{A}$ is equal to :-

1. $\frac{1}{2}$

2. 4

3. $\frac{3}{2}$

4. 2

Correct Option: , 4

Solution:

$\mathrm{Ax}_{1}=\mathrm{b}_{1}$

$\mathrm{Ax}_{2}=\mathrm{b}_{2}$

$\mathrm{Ax}_{3}=\mathrm{b}_{3}$

$\Rightarrow \quad|\mathrm{A}|\left|\begin{array}{lll}1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & 1\end{array}\right|=\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right|$

$\Rightarrow|\mathrm{A}|=\frac{4}{2}=2$