It&#x27;s a little more complicated than:
&quot;&quot;&quot;It&#x27;s funny because the whole discussion only exists because there is no division operator in &quot;matrix math&quot;.&quot;&quot;&quot;<p>Matrices in general (insert caveats on size here) form a ring, but not a division ring (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Division_ring" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Division_ring</a>).  This means that not all (square) matrices will have a multiplicative inverse.  This is why we often restrict to things like GL(n), the group of non-singular square matrices of size n.<p>Furthermore when you say &quot;scalars&quot;, they have multiplicative inverses because they&#x27;re elements of a field (and sometimes the inverse might be a pain like if you&#x27;re in a number field or something).<p>This isn&#x27;t even getting into the Moore-Penrose pseudoinverse (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Moore–Penrose_pseudoinverse" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Moore–Penrose_pseudoinverse</a>) or linear operators or so on, but I&#x27;ll let someone else shout about algebra.