I am only correcting the answer given by Sandeep_Uniyal
We can express this determinant as recurrence relation $|A_n|=3|A_{n-1}|-|A_{n-2}|$
Characteristic equation for this recurrence relation is $t^2-3t+1=0$
solving the equation we get,
$$ t = \frac{3\pm\sqrt{5}}{2} $$
Hence the solution for recurrence realtion is,
$$|A_n|=C_1t^n+C_2t^n$$
$$|A_n|=C_1\left(\frac{3\pm\sqrt{5}}{2}\right)^n+C_2\left(\frac{3\pm\sqrt{5}}{2}\right)^n$$
Looking at this solution, we can easily say only option D match the form of the equation.
So answer is option D.
But if you want complete solution, you have to dig some more.
solve the following two equation, which we got after putting $n=1,2$
$$|A_1|=3=C_1\left(\frac{3\pm\sqrt{5}}{2}\right)+C_2\left(\frac{3\pm\sqrt{5}}{2}\right)$$
$$|A_2|=8=C_1\left(\frac{7\pm3\sqrt{5}}{2}\right)+C_2\left(\frac{7\pm3\sqrt{5}}{2}\right)$$