Mathematics - Determinant Question with Solution | TestHub

Given system of linear equations are
$3x-2y+z=b$

$5x-8y+9z=3$

$2x+y+az=-1$

For system to have no solution, 

$∆=\left|\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right|=0$

$\Rightarrow 3\left(-8a-9\right)+2\left(5a-18\right)+1\left(21\right)=0$

$\Rightarrow a=-3$

Also, ${∆}_2=\left|\begin{array}{ccc}3& -2& b\\ 5& -8& 3\\ 2& 1& -1\end{array}\right|=3\left(5\right)+2\left(-11\right)+b\left(21\right)=21b-7$

If $b=\frac{1}{3}$, ${∆}_2=0$

So $b$ must be equal to $-\frac{1}{3}$