This question tests understanding of the distributive property for matrix multiplication over addition. The distributive property states that matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC, allowing us to expand or factor matrix expressions just like with numbers. To expand A(B + C), we apply the distributive property: A(B + C) = AB + AC, meaning we multiply A by B and A by C separately, then add the results, just as we would distribute a factor across a sum in algebra. Choice A is correct because it properly applies the distributive property to get A(B + C) = AB + AC. Choice D reverses the order in the expanded form, writing BA + CA instead of AB + AC, but order matters in matrix multiplication. The distributive property A(B + C) = AB + AC is crucial for expanding matrix expressions, and it works just like distributing with numbers, but remember to keep the multiplication order the same (A on the left in both terms).

This question tests understanding of identifying which property is illustrated by a given equation. The distributive property states that matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC, allowing us to expand or factor matrix expressions just like with numbers. The equation A(B + C) = AB + AC specifically shows how matrix multiplication distributes over addition from the left, meaning we can multiply A by the sum (B + C) or multiply A by each term separately and add the results. Choice C is correct because it correctly identifies this as the distributive property of matrix multiplication over addition. Choice A incorrectly identifies this as commutativity of multiplication, but commutativity would be AB = BA, not the distribution shown. To remember the properties: the distributive property A(B + C) = AB + AC shows how multiplication interacts with addition, while commutativity (AB = BA) and associativity ((AB)C = A(BC)) describe how operations behave with themselves.

This question tests understanding of special cases where matrix multiplication is commutative. Unlike multiplication of real numbers, matrix multiplication is not commutative: for most matrices A and B, AB ≠ BA, and even when both products are defined (square matrices), they typically give different results—only special cases like multiplication by the identity matrix commute. Matrix multiplication commutes in special cases: AB = BA when B is the identity matrix I (since AI = IA = A), when both are diagonal matrices with specific patterns, or when one is a scalar multiple of the identity, but these are exceptions, not the general rule. Choice C is correct because it correctly identifies a special case where AB = BA: when B = I (the identity matrix), since AI = IA = A. Choice A overgeneralizes from a special case, claiming AB = BA for all square matrices based on examples where it happens to be true (like with identity), but this doesn't hold generally. Special cases where AB = BA: when B is the identity matrix (AI = IA = A), when both matrices are scalar multiples of I, or when both are certain diagonal matrices, but don't assume commutativity generally.

This question tests understanding of the commutativity property for matrix multiplication. Matrix multiplication is generally NOT commutative (AB ≠ BA in general) even though it is associative ((AB)C = A(BC)). For example, with A = [[1,2],[3,4]] and B = [[0,1],[-1,2]], computing AB gives [[-2,5],[-4,11]] while BA gives [[3,4],[5,6]], demonstrating that the order of multiplication matters—reversing the order changes the outcome. Choice B is correct because it accurately states that matrix multiplication is not commutative in general, so typically AB ≠ BA. Choice A incorrectly claims that matrix multiplication is commutative for all square matrices, but this is false—only special matrices like the identity commute with all matrices. Key difference from numbers: matrix multiplication is NOT commutative (AB ≠ BA in general), so always pay attention to the order when multiplying matrices, but addition IS commutative (A + B = B + A), so order doesn't matter when adding.

This question tests understanding of the distributive property when multiplication occurs on the right. The distributive property states that matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC, allowing us to expand or factor matrix expressions just like with numbers. To expand (A + B)C, we apply the right distributive property: (A + B)C = AC + BC, meaning we multiply A by C and B by C separately, then add the results, maintaining the order with C on the right in both terms. Choice A is correct because it properly applies the distributive property to get (A + B)C = AC + BC. Choice B reverses the order in the expanded form, writing CA + CB instead of AC + BC, but order matters in matrix multiplication since it's not commutative. The distributive property works from both sides: A(B + C) = AB + AC (left distribution) and (A + B)C = AC + BC (right distribution), but remember to keep the multiplication order consistent in the expanded form.

This question tests understanding of the commutativity property for matrix multiplication. Unlike multiplication of real numbers, matrix multiplication is not commutative: for most matrices A and B, AB ≠ BA, and even when both products are defined (square matrices), they typically give different results—only special cases like multiplication by the identity matrix commute. For example, with A = [[1,2],[3,4]] and B = [[0,1],[-1,2]], computing AB gives [[−2,5],[−4,11]] while BA gives [[3,4],[5,6]], demonstrating that the order of multiplication matters—reversing the order changes the outcome. Choice B is correct because it accurately states that matrix multiplication is not commutative in general, so typically AB ≠ BA even when both products are defined. Choice A incorrectly claims that matrix multiplication is commutative for all square matrices, but this is false—commutativity only holds in special cases, not generally. Key difference from numbers: matrix multiplication is NOT commutative (AB ≠ BA in general), so always pay attention to the order when multiplying matrices, but addition IS commutative (A + B = B + A), so order doesn't matter when adding.

This question tests understanding of the distributive property for matrix operations. The distributive property states that matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC, allowing us to expand or factor matrix expressions just like with numbers. To expand A(B + C), we apply the distributive property: A(B + C) = AB + AC, meaning we multiply A by B and A by C separately, then add the results, just as we would distribute a factor across a sum in algebra. Choice B is correct because it properly applies the distributive property to get A(B + C) = AB + AC. Choice D reverses the order in the expanded form, writing BA + CA instead of AB + AC, but order matters in matrix multiplication. The distributive property A(B + C) = AB + AC is crucial for expanding matrix expressions, and it works just like distributing with numbers, but remember to keep the multiplication order the same (A on the left in both terms). 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.

This question tests understanding of scalar multiplication properties with matrix operations. 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). For scalar multiplication with matrices, the scalar can be associated with either matrix in a product: k(AB) = (kA)B = A(kB), which follows from the associative property and the fact that scalar multiplication commutes with matrix multiplication in this specific sense. Choice A is correct because it accurately states that k(AB) = (kA)B = A(kB), showing that the scalar k can be associated with either matrix in the product. Choice C incorrectly expands k(AB) as kA + kB, but scalar multiplication doesn't distribute over matrix multiplication like this—it associates with individual matrices in the product. 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. 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.

This question tests understanding of the non-commutativity of matrix multiplication. Unlike multiplication of real numbers, matrix multiplication is not commutative: for most matrices A and B, AB ≠ BA, and even when both products are defined (square matrices), they typically give different results—only special cases like multiplication by the identity matrix commute. For example, with A = [[1,2],[0,1]] and B = [[3,0],[4,3]], computing AB gives [[11,6],[4,3]] while BA gives [[3,6],[4,11]], demonstrating that the order of multiplication matters—reversing the order changes the outcome. Choice D is correct because it accurately states that matrix multiplication is not commutative in general, so AB ≠ BA is possible even for square matrices. Choice A incorrectly claims that all 2×2 matrices commute under multiplication, but this is false—commutativity only holds in special cases, not for all square matrices of the same size. Key difference from numbers: matrix multiplication is NOT commutative (AB ≠ BA in general), so always pay attention to the order when multiplying matrices, but addition IS commutative (A + B = B + A), so order doesn't matter when adding. 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.

This question tests understanding of the properties of matrix operations, specifically associativity of multiplication. Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)), but matrix multiplication is generally NOT commutative (AB ≠ BA in general) even though it is associative ((AB)C = A(BC)). 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.