Question

`\begin{gathered} \rm Find \: the \: cuberoot \: of \: the \: determinant \: given \: below. \\ \rm write \: the \: answer \: in \: the \: form \: (a)/(b), \: where \: a \: and \: b \: are \\ \rm positive \: integers \: and \: gcd (a,b) = 1.\end{gathered} \\ \begin{gathered}\left | \begin{matrix} \frac{ - {2019}^( 2 ) }{\sqrt[5]{80}} & \frac{ {2020}^(2) }{ \sqrt[5]{ {80}^(2) } }& \frac{ - {2021}^(2) }{ \sqrt[5]{ {80}^(3) } } \\ \\ \frac{ {2022}^(2) }{ \sqrt[5]{ {80}^(4) } } & \frac{ - {2023}^(2) }{ \sqrt[5]{ {80}^(5) } } & \frac{ {2024}^(2) }{ \sqrt[5]{ {80}^6 } } \\ \\ \frac{ - {2025}^(2) }{ \sqrt[5]{ {80}^(7) } }& \frac{ {2026}^(2) }{ \sqrt[5]{ {80}^(8) } } & \frac{ - {2027}^(2) }{ \sqrt[5]{ {80}^9 } } \end{matrix} \right | \end{gathered}`​

Answer 1

Quick warning that this is not the most elegant solution.

Let
`x=2019` and
`y=80^(1/5)`, so the matrix whose determinant we want looks like this:

`M = \begin{bmatrix} -\frac{x^2}y & ((x+1)^2)/(y^2) & -((x+2)^2)/(y^3) \\\\ ((x+3)^2)/(y^4) & -((x+4)^2)/(y^5) & ((x+5)^2)/(y^6) \\\\ -((x+6)^2)/(y^7) & ((x+7)^2)/(y^8) & -((x+8)^2)/(y^9) \end{bmatrix}`

Since it's a 3x3 matrix, we can just compute the determinant directly using a Laplace expansion. Along the first row, for instance, we get

`\det(M) = -\frac{x^2}y \begin{vmatrix} -((x+4)^2)/(y^5) & ((x+5)^2)/(y^6) \\\\ ((x+7)^2)/(y^8) & -((x+8)^2)/(y^9) \end{vmatrix} \\\\ - ((x+1)^2)/(y^2) \begin{vmatrix} ((x+3)^2)/(y^4) & ((x+5)^2)/(y^6) \\\\ -((x+6)^2)/(y^7) & -((x+8)^2)/(y^9) \end{vmatrix} \\\\ - ((x+2)^2)/(y^3) \begin{vmatrix} ((x+3)^2)/(y^4) & -((x+4)^2)/(y^5) \\\\ -((x+6)^2)/(y^7) & ((x+7)^2)/(y^8) \end{vmatrix}`

`\det(M) = -\frac{x^2}y \cdot ((x+4)^2(x+8)^2-(x+5)^2(x+7)^2)/(y^(14)) \\\\ - ((x+1)^2)/(y^2) \cdot ((x+5)^2(x+6)^2 - (x+3)^2(x+8)^2)/(y^(13)) \\\\ - ((x+2)^2)/(y^3) \cdot ((x+3)^2(x+7)^2 - (x+4)^2(x+6)^2)/(y^(12))`

`\det(M) = \frac{\text{numerator}}{y^(15)}`

where the numerator is

`\bigg((x+5)^2(x+7)^2 - (x+4)^2(x+8)^2\bigg)x^2 \\ + \bigg((x+3)^2(x+8)^2 - (x+5)^2(x+6)^2\bigg) (x+1)^2 \\ + \bigg((x+4)^2(x+6)^2 - (x+3)^2(x+7)^2\bigg) (x+2)^2`

Now just simplify. The numerator reduces drastically to a constant, 216. Then

`\det(M) = (216)/(\left(80^(1/5)\right)^(15)) = (6^3)/(80^3) = \left(\frac3{40}\right)^3 = \boxed{(27)/(64,000)}`