If ar = cos \(\frac{2 r \pi}{9}\) + i sin \(\frac{2 r \pi}{9}\), r = 1, 2, 3,...., i = √-1, then the determinant \(\begin{vmatrix} a_1 & a_2 & a_3 \\[0.3em] a_4 & a_5 & a_6 \\[0.3em] a_7 & a_8 & a_9 \end{vmatrix}\) is equal to:
(1) a2a6 - a4a8
(2) a9
(3) a1a9 - a3a7
(4) a5
Correct answer: (3) a1a9 - a3a7
Explanation:
ar = \(e^{\frac{i 2\pi r}{9}}\), r = 1, 2, 3,...., a1, a2, a3,.... are in G.P.
\(\begin{vmatrix} a_1 & a_2 & a_3 \\[0.3em] a_4 & a_5 & a_6 \\[0.3em] a_7 & a_8 & a_9 \end{vmatrix}\) = \(\begin{vmatrix} a_1 & a_2^2 & a_1^3 \\[0.3em] a_1^4 & a_1^5 & a_1^6 \\[0.3em] a_1^7 & a_1^8 & a_1^9 \end{vmatrix}\)
= a17.a14.a17
\(\begin{vmatrix} 1 & a_1 & a_1^2 \\[0.3em] 1 & a_1 & a_1^2 \\[0.3em] 1 & a_1 & a_1^2 \end{vmatrix}\)
Now a1a9 - a3a7 = a110 - a110 = 0