Summary

This chapter developed the algebra of matrices with real entries from first principles. A matrix was defined as a rectangular array $A=[a_{ij}]$ of order $m\times n$, and the standard shape- and pattern-based types were classified: row, column, square, zero, diagonal, scalar and identity matrices. Equality of matrices was defined by the two-part condition of matching order and matching corresponding entries, which converts a matrix equation into an ordinary system of equations to solve.

Addition (defined only for matrices of the same order) and scalar multiplication were shown to satisfy familiar laws -- commutativity, associativity, an additive identity and inverse, and distributivity -- exactly paralleling real-number arithmetic. Multiplication, by contrast, requires a compatibility condition on orders, is computed by the row-by-column rule $(AB)_{ij}=\sum_k a_{ik}b_{kj}$, and was shown, by concrete counterexample, to be non-commutative in general ($AB\neq BA$); a second worked example exhibited two non-zero $2\times2$ matrices whose product is nonetheless the zero matrix -- a zero-divisor phenomenon with no real-number analogue, and the reason matrix "cancellation" is unsafe without knowing a factor is invertible.

The transpose $A^T$ (rows and columns interchanged) satisfies $(A^T)^T=A$, $(A+B)^T=A^T+B^T$, $(kA)^T=kA^T$, and the reversal law $(AB)^T=B^TA^T$. Building on transpose, a square matrix was called symmetric if $A^T=A$ and skew-symmetric if $A^T=-A$ (forcing every diagonal entry to be zero); every square matrix was proved to split uniquely into a symmetric part $\tfrac12(A+A^T)$ and a skew-symmetric part $\tfrac12(A-A^T)$. …