#Type/Theorem #Topic/Linear_Algebra

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 9.4 Matrix-Matrix Addition, 9.5 Scalar-Matrix Multiplication

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Just like regular arithmetic operations, there are a number of algebraic properties for matrix operations.

Algebraic Properties of Matrix Operations

Let matrices A,B,C,∈Rm×n and scalars a,b∈R. Then, all of the following are properties of matrix addition:

  1. Commutativity of matrix addition: A+B=B+A
  2. Associativity of matrix: A+(B+C)=(A+B)+C
  3. Additive Identity: A+0=0+A=A
  4. Additive Inverses: A+−A=−A+A=0
  5. Distributivity of matrix addition: a(A+B)=aA+aB
  6. Distributivity of scalar addition: (a+b)A=aA+bA
  7. Associativity of scalar multiplication: a(bA)=(ab)A
  8. Multiplicative Identity of scalar multiplication: 1A=A

Remark. Notice how the properties of matrix addition and scalar-matrix multiplication are the exact same as the 4.3 Algebraic Properties of Vector Arithmetic.
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