Matrices and Matrix Operations

Matrices and Basic Operations

A matrix is a rectangular array of numbers arranged in rows and columns. An $m \times n$ matrix has $m$ rows and $n$ columns. They compactly represent linear systems, transformations, and data tables.

The identity matrix $I$ has ones on the diagonal and zeros elsewhere, acting like the number 1 so that $AI = A$. A square matrix may have an inverse $A^{-1}$ where $AA^{-1} = I$, though not all matrices are invertible.

Matrix Multiplication & Transformations

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. An $m \times n$ times an $n \times p$ matrix yields an $m \times p$ result. Entry $(i, j)$ is the dot product of row $i$ of $A$ and column $j$ of $B$.

Matrix multiplication is not commutative: in general $AB \neq BA$, though it is associative $A(BC) = (AB)C$. This odd rule matters because a matrix represents a linear transformation, and multiplying matrices composes those transformations in sequence.

Common pitfall: Never multiply matrices entry-by-entry. Always use the dot product of a row and a column, and verify that shapes match.