Matrix reference
An \(m\times n\) matrix has \(m\) rows and \(n\) columns:
\[A=(a_{ij}).\]Multiplication
If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then \(AB\) is \(m\times p\), with
\[(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.\]In general, \(AB\ne BA\).
Identity
\[AI=IA=A.\]Eigenvalues
\[A\mathbf v=\lambda\mathbf v,\qquad \mathbf v\ne0.\]Eigenvalues appear in stability, population projection, networks and Markov models.
Linear systems
\[A\mathbf x=\mathbf b.\]When solving numerically, direct linear-system solvers are usually preferable to explicitly forming \(A^{-1}\).