\section{Matrici}
$A_{m,n} \cdot B_{n,p} = C_{m,p}$
$C_{i,j} = \sum_{k=1}^{n} a_{ik} b_{kj}$ ($i$-esima riga di A $\cdot$ $j$-esima colonna di B)
\begin{tabular}{ll}
Non commutativa & $A B \ne B A$ \\
Associativa & $A(BC) = (AB)C$ \\
Elemento neutro & $AI_n = I_nA = A$ \\
Distributiva & $A(B+C) = AB + AC$ \\
Trasposta & $A^T = [\alpha_{ij}] \quad \alpha_{ij} = a_{ji}$ \\
Inversa & $AA^{-1} = A^{-1}A = I_n\quad (\exists\ \text{sse} \det A \ne 0)$ \\
& $(AB)^{-1} = B^{-1}A^{-1}$ \\
\end{tabular}
$[A|I_n] \rightarrow [I_n|A^{-1}]$ riducendo a scala
$A^{-1} = (\nicefrac{1}{\det{A}})\ \left[b_{ik}\right] \quad b_{ik} = C_{ki}$ (compl. algebrico trasposto)
$A =
\left[\begin{array}{c|c|c|c} \vec{C}_1 & \vec{C}_2 & \cdots & \vec{C}_n \end{array}\right]
=
\left[\arraycolsep=1.7pt\def\arraystretch{1}\begin{array}{c} \vec{R}_1 \\ \hline \vdots \\ \hline \vec{R}_m \end{array}\right]$
$\Row A = \Lin(\vec{R}_1, \vec{R}_2, ..., \vec{R}_m) \quad \Col A = \Lin(\vec{C}_1, \vec{C}_2, ..., \vec{C}_n)$
$\dim \Row A = \dim \Col A = \rk A$
