Solution
= Solution
The update must place the sum of the two current entries in the first coordinate and retain the old first coordinate in the second. Thus choose the <matrix>
$$
\boxed{A=\begin{pmatrix}1&1\\1&0\end{pmatrix}.}
$$
By <matrix multiplication>, $A(L_n,L_{n-1})^T=(L_n+L_{n-1},L_n)^T=(L_{n+1},L_n)^T$ for every $n\geq1$, using the given <recurrence relation>. This is the companion-matrix representation of the <Lucas numbers>.