Which matrix operation is NOT commutative?a.Matrix inversionb.Matrix additionc.Matrix multiplicationd.Matrix subtraction

1. Break Down the Problem

We need to identify which matrix operation listed—matrix inversion, addition, multiplication, or subtraction—is not commutative.

2. Relevant Concepts

• Commutative Property: An operation is commutative if changing the order of the operands does not change the result.
  • For addition: $A + B = B + A$
  • For multiplication: $A \times B = B \times A$
  • For subtraction: $A - B \neq B - A$ (generally)
  • For matrix inversion, $A^{-1}$ does not apply in the same sense of order as the other operations.

3. Analysis and Detail

1. Matrix Addition: It is commutative, meaning $A + B = B + A$.
2. Matrix Multiplication: It is generally not commutative, meaning $A \times B \neq B \times A$ in most cases.
3. Matrix Subtraction: It is not commutative; $A - B \neq B - A$.
4. Matrix Inversion: While it follows the associative property in terms of the operation $AA^{-1} = I$, it does not fit into the commutative operations like addition or multiplication, since the product doesn’t apply here.

4. Verify and Summarize

Since both matrix multiplication and subtraction exhibit non-commutative behavior, it is crucial to note that when asked for a singular answer concerning operations, multiplication is most commonly highlighted in mathematical literature as the primary example of a non-commutative operation.

Final Answer

Thus, the correct answer is c. Matrix multiplication as the operation that is NOT commutative.