Suppose the vectors x1, x2 and x3 are the solutions

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

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

$b_{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right]$, then the determinant of $A$ is equal to :

  1. (1) 4

  2. (2) 2

  3. (3) $\frac{1}{2}$

  4. (4) $\frac{3}{2}$


Correct Option: , 2

Solution:

Given that $A x=b$ has solutions $x_{1}, x_{2}, x_{3}$ and $b$ is

equal to $b_{1}, b_{2}$ and $b_{3}$

$\therefore x_{1}+y_{1}+z_{1}=1$

$\Rightarrow 2 y_{1}+z_{1}=2 \Rightarrow z_{1}=2$

Determinant of coefficient matrix

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

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