Precalculus Question of the Day

1. No; matrix multiplication is not associative, so $(AB)C\ne A(BC)$ in general.
2. Yes; matrix multiplication is associative, so $(AB)C=A(BC)$ whenever both sides are defined. (correct answer)
3. Yes, but only when $AB=BA$ (commutativity is required for associativity).
4. No; $(AB)C=A(BC)$ holds only when $A$ is the identity matrix $I$.

Explanation: This question tests understanding of the properties of matrix operations, specifically the associativity of matrix multiplication. Matrix multiplication has the associative property, meaning (AB)C = A(BC), so we can group operations differently without changing the result, but it lacks commutativity, so we cannot change the order of multiplication (AB ≠ BA in general). The associative property allows us to compute (AB)C or A(BC) and get the same result, so we can choose which multiplication to perform first based on computational convenience without worrying about different answers. Choice B is correct because it correctly describes associativity. Choice A incorrectly claims that matrix multiplication is not associative, but (AB)C = A(BC) always holds (when dimensions are compatible). When simplifying matrix expressions, you can freely use associativity to regroup operations ((AB)C = A(BC)) and distributivity to expand or factor (A(B + C) = AB + AC), but you cannot change the order of multiplication without potentially changing the result. To remember the properties: addition and multiplication are both associative (can regroup) and both operations distribute over addition, but only addition is commutative (can reorder)—multiplication order matters.