https://www.chenk.top/en/tags/signed-volume/Recent content in Signed Volume on Chen Kai BlogHugoenWed, 22 Jan 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/04-the-secrets-of-determinants/Wed, 22 Jan 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/04-the-secrets-of-determinants/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/linear-algebra/04-the-secrets-of-determinants/illustration_1.png" alt="Essence of Linear Algebra (4): The Secrets of Determinants — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="beyond-the-formula" class="heading-anchor">Beyond the Formula<a href="#beyond-the-formula" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><span class="math-block">$$\det\begin{pmatrix}a &amp; b\\ c &amp; d\end{pmatrix} = ad - bc$$</span> <p> You plug in numbers, compute, and move on. That misses the point entirely.</p> <p>Here is the real meaning, in one sentence:</p> <blockquote> <p><strong>The determinant of <span class="math-inline">$A$</span> is the factor by which <span class="math-inline">$A$</span> scales area (in 2D) or volume (in 3D).</strong></p>