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.

Properties of Matrix Arithmetic

We will now outline some important properties of matrix addition, scalar multiplication, matrix multiplication, matrix transposes, and matrix inverses in the following theorem.

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 $k \neq 0$.

The proofs of the statements given above can be obtained by using the definitions of the operations on matrices directly. Many of them are relatively trivial and can be found on the Linear Algebra section of Math Online for those who are interested.

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