Final answer:
The properties in question relate to matrix operations in mathematics, specifically stating that matrix multiplication is associative and distributive with respect to scalar multiplication and unaffected by the identity matrix, while vector addition is commutative and associative.
Step-by-step explanation:
The properties discussed pertain to matrix and vector operations within mathematics, specifically linear algebra and vector calculus. The Commutative property for matrices (AB = BA) generally does not hold true, unlike for scalar numbers, because matrix multiplication is not commutative. The Associative property (A(BC) = (AB)C), however, does hold for matrices, which means that the way matrices are grouped during multiplication doesn't change the result. For Scalar Multiplication, the property c(AB) = (cA)B = A(cB) indicates that a scalar can multiply matrices in any order, and the result will be the same. Finally, the Identity Matrix property (AI = A) shows that when any matrix (A) is multiplied by an identity matrix (I), the original matrix (A) is unchanged.
Concerning vector operations, scalar multiplication is distributive and vector addition is both commutative and associative, meaning vectors can be added in any order and multiplied by scalars which can then be distributed over vector addition. This is critical when calculating properties like dot and cross products in physics and engineering disciplines.