The values of α , for which | cases 1 & (3)/(2) & α + (3)/(2) 1 &...

Given: \left|\begin{cases}1 & \dfrac{3}{2} & \alpha + \dfrac{3}{2} \\ 1 & \dfrac{1}{3} & \alpha + \dfrac{1}{3} \\ 2\alpha + 3 & 3\alpha + 1 & 0\end{cases}\right| = 0

Applying, ${R}_{1} \to {R}_{1} - {R}_{2}$

⇒ \left|\begin{cases}0 & \dfrac{7}{6} & \dfrac{7}{6} \\ 1 & \dfrac{1}{3} & \alpha + \dfrac{1}{3} \\ 2\alpha + 3 & 3\alpha + 1 & 0\end{cases}\right| = 0

Applying, ${C}_{2} \to {C}_{2} - {C}_{3}$

⇒ \left|\begin{cases}0 & 0 & \dfrac{7}{6} \\ 1 & - \alpha & \alpha + \dfrac{1}{3} \\ 2\alpha + 3 & 3\alpha + 1 & 0\end{cases}\right| = 0

$⇒ 0 - 0 + \dfrac{7}{6}\left(3\alpha + 1 + 2{\alpha}^{2} + 3\alpha\right) = 0$

$⇒ 2{\alpha}^{2} + 6\alpha + 1 = 0$

$⇒ \alpha = \dfrac{- 6 \pm \sqrt{36 - 4\left(2\right)\left(1\right)}}{2 \times 2}$

$⇒ \alpha = \dfrac{- 6 \pm 2\sqrt{7}}{4}$

$⇒ \alpha = \dfrac{- 3 \pm \sqrt{7}}{2}$

$⇒ \alpha \approx \dfrac{- 3 \pm 2 . 6}{2}$

$⇒ \alpha \approx \dfrac{- 0 . 4}{2}, \dfrac{- 5 . 6}{2}$

$⇒ \alpha \approx - 0 . 2, - 2 . 8$

$⇒ \alpha \in \left(- 3, 0\right)$