Question

Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear - equations, $Ax = b$ when the vector $b$ on the right side is equal to $b _{1}, b _{2}$ and $b _{3}$ respectively. If
$X=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix},X_{2}\begin{bmatrix}0\\ 2\\ 1\end{bmatrix},X_{3}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},b_{1}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}$
$b_{2}=\begin{bmatrix}0\\ 2\\ 0\end{bmatrix}$ and $b_{3}= \begin{bmatrix}0\\ 0\\ 2\end{bmatrix}$,then the determinant of $A$ is equal to :

• A. $\frac{1}{2}$
• B. 4
• C. $\frac{3}{2}$
• D. 2

Answer 1

Answer: (D)

$Ax _{1}= b _{1}$
$Ax _{2}= b _{2}$
$Ax _{3}= b _{3}$
$\Rightarrow \left|A\right|\begin{vmatrix}1&0&0\\ 1&2&0\\ 1&1&1\end{vmatrix}=\begin{vmatrix}1&0&0\\ 0&2&0\\ 0&0&2\end{vmatrix}$
$\Rightarrow | A |=\frac{4}{2}=2$