Mathematics - Determinant Question with Solution | TestHub

Given system of linear equations are $x+y-z=2,x+2y+\alpha z=1,2x-y+z=\beta$

Using Cramer Rule, we can say for infinite solutions

$\Delta ={\Delta }_1={\Delta }_2={\Delta }_3=0$

Here, $\Delta =\left|\begin{array}{ccc}1& 1& -1\\ 1& 2& \alpha \\ 2& -1& 1\end{array}\right|=0$

Apply $R_1\leftrightarrow R_1+R_3$

$\Rightarrow \left|\begin{array}{ccc}3& 0& 0\\ 1& 2& \alpha \\ 2& -1& 1\end{array}\right|=0$

$\Rightarrow 3\left(2+\alpha \right)=0$

$\Rightarrow \alpha =-2$

And ${\Delta }_2=\left|\begin{array}{ccc}1& 2& -1\\ 1& 1& -2\\ 2& \beta & 1\end{array}\right|=0$

$\Rightarrow 1\left(1+2\beta \right)-2\left(1+4\right)-\left(\beta -2\right)=0$

$\Rightarrow \beta -7=0$

$\Rightarrow \beta =7$

$\therefore \alpha +\beta =5$