Mathematics · Ch 3 — Matrices

Summary

A matrix is a rectangular array of numbers arranged in $m$ rows and $n$ columns, denoted as $A = [a_{ij}]_{m \times n}$ .
Order of a matrix is $m \times n$ ; two matrices are equal only if they have the same order and corresponding elements are equal.
Types of matrices: row ( $1 \times n$ ), column ( $m \times 1$ ), square ( $m = n$ ), diagonal (non-diagonal entries zero), scalar (diagonal with equal entries), identity ( $I_n$ ), zero ( $O$ ).
Addition/Subtraction: Only for matrices of same order; done element-wise: $(A \pm B)_{ij} = a_{ij} \pm b_{ij}$ .
Scalar multiplication: $kA = [k \cdot a_{ij}]$ ; matrix multiplication $A_{m \times n} \times B_{n \times p}$ gives $C_{m \times p}$ where $c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$ .
Properties: Matrix multiplication is associative ( $A(BC) = (AB)C$ ), distributive ( $A(B+C) = AB + AC$ ), but not commutative ( $AB \neq BA$ generally).
Transpose $A'$ (or $A^T$ ): rows become columns; $(A')' = A$ , $(A+B)' = A' + B'$ , $(kA)' = kA'$ , $(AB)' = B'A'$ .
Symmetric matrix: $A' = A$ (for square matrices); Skew-symmetric: $A' = -A$ (diagonal entries are zero).
Elementary row/column operations: Interchange ( $R_i \leftrightarrow R_j$ ), multiply by non-zero scalar ( $R_i \to kR_i$ ), add multiple of another row ( $R_i \to R_i + kR_j$ ). …