Q. If $a_{1} a_{2} a_{3} ...... a_{9}$ are in A.P. then the value of
$∣ ∣ a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9} ∣ ∣$ is

4185 161 KCETKCET 2020Determinants Report Error

A

$\frac{9}{2} (a_{1} + a_{9})$

28%

B

$(a_{1} + a_{9})$

22%

C

$l o g_{e} (l o g_{e} e)$

21%

D

1

29%

Solution:

Let $d$ be the common difference of A.P.
Then, $Δ = ∣ ∣ a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9} ∣ ∣ = ∣ ∣ a_{1} a_{1} + 3 d a_{1} + 6 d a_{1} + d a_{1} + 4 d a_{1} + 7 d a_{1} + 2 d a_{1} + 5 d a_{1} + 8 d ∣ ∣$ Applying $C_{2} \to C_{2} - C_{1}$ and $C_{3} \to C_{3} - C_{2}$
$Δ = ∣ ∣ a_{1} a_{1} + 3 d a_{1} + 6 d d d d d d d ∣ ∣$
$= 0$
$(∵ C_{2}$ and $C_{3}$ are identical)
$= lo g_{e} (lo g_{e} e) (∵ lo g_{e} 1 = 0)$