Matrix

A matrix is a rectangular array of entries (typically from a ring or field ) arranged in rows and columns.

An $m \times n$ matrix $A$ has $m$ rows and $n$ columns, written

The entry in row $i$ and column $j$ is denoted $a_{ij}$ or $A_{ij}$.

Operations

• Addition: $(A + B)_{ij} = A_{ij} + B_{ij}$ (requires same dimensions).
• Scalar multiplication: $(cA)_{ij} = c \cdot A_{ij}$.
• Matrix multiplication: $(AB)_{ij} = \sum_k A_{ik} B_{kj}$ (requires compatible dimensions).

Special matrices

• Square matrix: $m = n$.
• Identity matrix: $I_n$ with $1$s on diagonal, $0$s elsewhere.
• Zero matrix: All entries are $0$.
• Diagonal matrix: Non-zero entries only on the main diagonal.
• Symmetric matrix: $A = A^T$ (equals its transpose).

Matrices represent linear maps between finite-dimensional vector spaces once bases are chosen.