# If $a_1, \ldots ,a_9$ are in harmonic progression, then find the value of the determinant

If $a_1, \ldots ,a_9$ are in harmonic progression ,then find the value of the determinant $$\begin{vmatrix} a_1 &a_2&a_3\\ 5&4&a_6\\ a_7&a_8&a_9\\ \end{vmatrix}$$

I calculated the terms as $a_1=\frac{20}{1},a_2=\frac{20}{2},a_3=\frac{20}{3},a_4=\frac{20}{4},a_5=\frac{20}{5},a_6=\frac{20}{6},a_7=\frac{20}{7},a_8=\frac{20}{8},a_9=\frac{20}{9}$ but now calculating the determinant is tedious.Is there some other elegant method possible?

• perform a raw or column operation – Deepesh Meena Aug 9 '18 at 15:53
• After taking out the factor $20$ in each column I fed it to Alpha and got $1/3360$. As it is non-zero I think it is tedious. Your determinant is then $20^3/3360$ – Ross Millikan Aug 9 '18 at 15:57
• This is a duplicate of math.stackexchange.com/questions/1755160/… which did not attract an answer either. – Ross Millikan Aug 9 '18 at 16:02

after taking $20^3$ common and performing the following row operations
replace $R_2$ with $R_2-\frac{R1}{4}$ replace $R_3$ with $R_3-\frac{R1}{7}$ I got this \begin{vmatrix} 1 & 1/2& 1/3\\ 0 & 3/40& 1/12\\ 0 & 3/56& 4/63\\ \end{vmatrix}
I got $det(A)=\frac{20^3}{3360}$