The Fibonacci sequence is defined by \(F_0 = 0,\), \(F_1 = 1,\), \(F_n = F_{n - 1} + F_{n - 2}\) for all \(n \ge 2.\)
\(\det \begin{pmatrix} F_{1000} & F_{1001} & F_{1002} \\ F_{1001} & F_{1002} & F_{1003} \\ F_{1002} & F_{1003} & F_{1004} \end{pmatrix} .\) (its not -4)
+9519 Note that the third row is just the first row plus the second row.
Doing row operation yields \(\det \begin{pmatrix} F_{1000} & F_{1001} & F_{1002} \\ F_{1001} & F_{1002} & F_{1003} \\ F_{1002} & F_{1003} & F_{1004} \end{pmatrix}=\det \begin{pmatrix} F_{1000} & F_{1001} & F_{1002} \\ F_{1001} & F_{1002} & F_{1003} \\ 0&0&0 \end{pmatrix}\).
The answer is now obvious if you are familiar with the properties of determinant.
