Final answer:
To determine the order of multiplying three matrices in a word problem, you must consider Based on the dimensions of the matrices. The number of columns in one matrix must match the number of rows in the next. This is crucial as matrix multiplication doesn't follow the commutative property and only multiplication that adheres to dimensional compatibility is possible.
Step-by-step explanation:
The order in which you multiply three matrices in a word problem is determined by C) Based on the dimensions of the matrices. This means that the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible. Therefore, despite the associative property of matrix multiplication, which allows us to multiply in any grouped order, the actual sequence of multiplication is dictated by the dimensions of the matrices involved. This is not to be confused with the commutative property, which does not hold for matrix multiplication, meaning that AB ≠ BA in most cases.
Multiplication should follow the correct order that agrees with these dimensions, otherwise, matrix multiplication will not be possible. In vector operations, addition follows the commutative property, e.g., A + B = B + A. However, it should be noted that this does not apply to matrix multiplication. Furthermore, unlike vector addition, the order of matrix multiplication can impact the result due to these dimensional constraints.
It is also important to distinguish between different types of vector multiplication when discussing properties such as commutative, associative, and distributive. For instance, the distributive property applies to both the dot (scalar) and cross (vector) product of vectors, but the commutative property applies only to the scalar product, while the cross product is anticommutative.