Let \(\left\{ {{f_0},\;{f_1},\; \ldots ,\;{f_n}} \right\}\) be the
set of \( n + 1\) Fibonacci numbers where \({f_0} = {f_1} = 1\)
and \({f_j} = {f_{j - 1}} + {f_{j - 2}},\quad 2 \le j \le n\). The
order \(n + 1\) Fibonacci matrix \({\bf{F}}\) has as typical element
\({F_{i,j}} = \left\{ {\begin{array}{cc}
{{f_{i - j + 1}}}&{i - j + 1 \ge 0}\\
0&{i - j + 1 < 0}
\end{array}} \right.\).