Question

Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent.
Let $|M|$ represent the determinant of the matrix $M=\left[\begin{array}{ccc}\alpha & 2& \gamma \\ \beta & 1& 0\\ -1& 0& 1\end{array}\right].$
Then the value of $|M|$ is

Answer 1

$\Delta =\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|$
$R_3\to R_3-R_1,R_2\to R_2-R_1$
$\Rightarrow \Delta =\left|\begin{array}{ccc}1& 2& 3\\ 3& 3& 3\\ 6& 6& 6\end{array}\right|$
$\Rightarrow \Delta =0$
Given system of equation will be consistent even if $\alpha =\beta =\gamma -1=0,$
i.e., equations will form homogeneous
system.
So, $\alpha =0,\beta =0,\gamma =1$
$\therefore |M|=\left|\begin{array}{ccc}0& 2& 1\\ 0& 1& 0\\ -1& 0& 1\end{array}\right|$
$\Rightarrow |M|=1$