Evaluating Matrix Properties Statements

This solution analyzes four statements regarding the properties of matrix addition and multiplication to determine which are correct.

Statement Analysis

• Statement A: Matrix Addition is not Commutative
  • The Commutative property for addition states that for matrices A and B, $ A + B = B + A $.
  • Matrix addition is indeed commutative, provided that matrices A and B have the same dimensions.
  • Therefore, Statement A is False.
• Statement B: Matrix Addition is Associative
  • The Associative property for addition states that for matrices A, B, and C, $ (A + B) + C = A + (B + C) $.
  • Matrix addition satisfies this property when the dimensions are compatible.
  • Therefore, Statement B is True.
• Statement C: Matrix Multiplication is Commutative
  • The Commutative property for multiplication states $ A \times B = B \times A $.
  • Matrix multiplication is generally not commutative. That is, $ A \times B $ is usually not equal to $ B \times A $, even when both products exist.
  • Therefore, Statement C is False.
• Statement D: Matrix Multiplication is Associative if product exists
  • The Associative property for multiplication states $ (A \times B) \times C = A \times (B \times C) $.
  • Matrix multiplication is associative, provided the dimensions of the matrices allow the products to be defined.
  • Therefore, Statement D is True.

Conclusion

Based on the analysis:

• Statement B is True.
• Statement D is True.

The correct option includes only the true statements, which are B and D.

Correct Answer: Option 4 (B and D only)