Question

Which of the following is equal to $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \theta & \varphi & \Psi \\ \lambda & \mu & \nu \end{array}\right|$ ?

• A. $\left|\begin{array}{ccc}\beta & \mu & \varphi \\ \alpha & \lambda & \theta \\ \gamma & \nu & \Psi \end{array}\right|$
• B. $-\left|\begin{array}{ccc}\beta & \mu & \varphi \\ \alpha & \lambda & \theta \\ \gamma & \nu & \Psi \end{array}\right|$
• C. $\left|\begin{array}{ccc}\alpha & \lambda & \theta \\ \beta & \mu & \varphi \\ \gamma & \nu & \Psi \end{array}\right|$
• D. $-\left|\begin{array}{ccc}\lambda & \alpha & \theta \\ \mu & \beta & \varphi \\ \nu & \gamma & \Psi \end{array}\right|$

Answer 1

The correct option is A $\left|\begin{array}{ccc}\beta & \mu & \varphi \\ \alpha & \lambda & \theta \\ \gamma & \nu & \Psi \end{array}\right|$

Let $\Delta =\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \theta & \varphi & \Psi \\ \lambda & \mu & \nu \end{array}\right|$
$=\left|\begin{array}{ccc}\alpha & \theta & \lambda \\ \beta & \varphi & \mu \\ \gamma & \Psi & \nu \end{array}\right|$
$=(-1)\left|\begin{array}{ccc}\alpha & \lambda & \theta \\ \beta & \mu & \varphi \\ \gamma & \nu & \Psi \end{array}\right|$ ------$(C_2\leftrightarrow C_3)$
$=(-1{)}^2\left|\begin{array}{ccc}\beta & \mu & \varphi \\ \alpha & \lambda & \theta \\ \gamma & \nu & \Psi \end{array}\right|$ -------$(R_1\leftrightarrow R_2)$
$=\left|\begin{array}{ccc}\beta & \mu & \varphi \\ \alpha & \gamma & \theta \\ \gamma & \nu & \Psi \end{array}\right|$