Let $$A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]$$ and $$B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right],\,\alpha \in C$$. Then the absolute value of the sum of all values of $\alpha$ for which det(AB) = 0 is :

<p>Given,</p> <p>$$A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]$$</p> <p>and $$B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right]$$</p> <p>$$AB = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]\left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right]$$</p> <p>$$ = \left[ {\matrix{ {4 + 4\alpha } & { - 4\alpha - 4} \cr {2\alpha - 6} & {{\alpha ^2} + 9} \cr } } \right]$$</p> <p>Given,</p> <p>$|AB| = 0$</p> <p>$\therefore$ $$\left| {\matrix{ {4 + 4\alpha } & { - 4\alpha - 4} \cr {2\alpha - 6} & {{\alpha ^2} + 9} \cr } } \right| = 0$$</p> <p>$$ \Rightarrow (4\alpha + 4)\left| {\matrix{ 1 & { - 1} \cr {2\alpha - 6} & {{\alpha ^2} + 9} \cr } } \right| = 0$$</p> <p>$\Rightarrow (4\alpha + 4)({\alpha ^2} + 9 + 2\alpha - 6) = 0$</p> <p>$\Rightarrow (4\alpha + 4)({\alpha ^2} + 2\alpha + 3) = 0$</p> <p>$\therefore$ $\alpha - = - 1$</p> <p>or ${\alpha ^2} + 2\alpha + 3 = 0$</p> <p>${\alpha _1} + {\alpha _2} = - 2$</p> <p>$\therefore$ Sum of all values of $\alpha = - 1 - 2 = - 3$</p> <p>$\therefore$ Absolute value of $\alpha = | - 3| = 3$</p>