Mathematics - Determinant Question with Solution | TestHub

$x+2y+3z=\alpha$

$4x+5y+6z=\beta$

$7x+8y+9z=\gamma -1$

The system of linear equations is consistent it means it has unique solution or infinite solutions

Here, $\Delta =\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|=0$

Hence, there are infinitely many solutions.

If equations have infinitely many solutions, then the equations are linearly connected, i.e., $L_1+\lambda L_2=L_3$

$\Rightarrow \left(x+2y+3z-\alpha \right)+\lambda (4x+5y+6z-\beta )=7x+8y+9z-\gamma +1$

$\frac{1+4\lambda }{7}=\frac{2+5\lambda }{8}=\frac{3+6\lambda }{9}=\frac{\alpha +\lambda \beta }{\gamma -1}$

$\frac{1+4\lambda }{7}=\frac{2+5\lambda }{8}\Rightarrow \lambda =-2$

Also,$\frac{1+4\lambda }{7}=\frac{\alpha +\lambda B}{\gamma -1}$

$\Rightarrow -1=\frac{\alpha -2\beta }{r-1}$

$\Rightarrow \alpha -2\beta +\gamma =1$

Now, $\left|M\right|=\left|\begin{array}{ccc}\alpha & 2& \gamma \\ \beta & 1& 0\\ -1& 0& 1\end{array}\right|$

$=\alpha +2(-\beta )+\gamma \left(1\right)=\alpha -2\beta +\gamma =1$