Question

Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equation
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent. Let $|M|$ represent the determinant of the matrix
$M=\left[\begin{array}{ccc}\alpha & 2& \gamma \\ \beta & 1& 0\\ -1& 0& 1\end{array}\right]$
Let $P$ be the plane containing all those ( $\alpha ,\beta ,\gamma$ ) for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
The value of $|M|$ is ________.

Answer 1

Answer: 1

$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
Equation $(1)+(3)-(2)=0$. Equation $(2)$ provides
$\alpha +\gamma -1-2\beta =0$
$|M|=\left|\begin{array}{ccc}\alpha & 2& \gamma \\ \beta & 1& 0\\ -1& 0& 1\end{array}\right|\left(C_1\to C_1+C_3\right)$
$=\left|\begin{array}{ccc}\alpha +\gamma & 2& \gamma \\ \beta & 1& 0\\ 0& 0& 1\end{array}\right|\left(R_1\to R_1-2R_2\right)$
$\left|\begin{array}{ccc}\alpha +\gamma -2\beta & 0& \gamma \\ \beta & 1& 0\\ 0& 0& 1\end{array}\right|=\left|\begin{array}{ccc}1& 0& \gamma \\ \beta & 1& 0\\ 0& 0& 1\end{array}\right|=1$