cp-algorithms
diff --git a/‎main/algebra/fibonacci-numbers.html
Lines changed: 13 additions & 3 deletions b/‎main/algebra/fibonacci-numbers.html
Lines changed: 13 additions & 3 deletions
@@ -6772,7 +6772,7 @@
 <ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
 
   Last update:
-  <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 22, 2025 01:19:59">April 22, 2025</span>&emsp;
+  <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 22, 2025 03:58:03">April 22, 2025</span>&emsp;
 
 <!-- Tags -->
 
@@ -6929,7 +6929,17 @@ <h3 id="matrix-form">Matrix form<a class="headerlink" href="#matrix-form" title=
 F_{n}
 \end{pmatrix}
 $$</div>
-<p>where <span class="arithmatex">$F_1 = 1, F_0 = 0$</span>.</p>
+<p>where <span class="arithmatex">$F_1 = 1, F_0 = 0$</span>. 
+In fact, since </p>
+<div class="arithmatex">$$
+\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}
+= \begin{pmatrix} F_2 &amp; F_1 \\ F_1 &amp; F_0 \end{pmatrix}
+$$</div>
+<p>we can use the matrix directly:</p>
+<div class="arithmatex">$$
+\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}^n
+= \begin{pmatrix} F_{n+1} &amp; F_n \\ F_n &amp; F_{n-1} \end{pmatrix}
+$$</div>
 <p>Thus, in order to find <span class="arithmatex">$F_n$</span> in <span class="arithmatex">$O(\log  n)$</span> time, we must raise the matrix to n. (See <a href="binary-exp.html">Binary exponentiation</a>)</p>
 <div class="highlight"><pre><span></span><code><span class="k">struct</span><span class="w"> </span><span class="nc">matrix</span><span class="w"> </span><span class="p">{</span>
 <span class="w">    </span><span class="kt">long</span><span class="w"> </span><span class="kt">long</span><span class="w"> </span><span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
@@ -7034,7 +7044,7 @@ <h2 id="practice-problems">Practice Problems<a class="headerlink" href="#practic
 
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