Properties of Matrix Arithmetic

This page is intended to be a part of the Numerical Analysis section of Math Online. Similar topics can also be found in the Linear Algebra section of the site.

 Theorem 1: Let $A$, $B$, and $C$ be matrices and let $k$ be a constant.. Then whenever the operations of matrix addition and matrix multiplication are defined, we have that: a) $A + (B + C) = (A + B) + C$ (Associativity of Matrix Addition). b) $A(BC) = (AB)C$ (Associativity of Matrix Multiplication). c) $A + B = B + A$ (Commutativity of Matrix Addition). d) $(A + B)C = AC + BC$ (Distributivity of Matrix Addition over Matrix Multiplication). e) $A(B + C) = AB + AC$ (Distributivity of Matrix Multiplication over Matrix Addition). f) $k(A + B) = kA + kB$. (Distributivity of Scalar Multiplication over Matrix Addition). g) $k(AB) = (kA)B = A(kB)$. h) $(A + B)^T = A^T + B^T$ (Distributivity of Transpose over Matrix Addition). i) $(AB)^T = B^TA^T$. j) $(AB)^{-1} = B^{-1} A^{-1}$. k) $(kA)^{-1} = \frac{1}{k} A^{-1}$ for $k \neq 0$.