1. Statement of the problem. Determine which one of the following properties does not hold for matrix multiplication. The options are: A. $A(BC)=(AB)C$. B. $A(B+C)=AB+AC$. C. $AI=A$. D. $BA=AB$. E. None. 2. Relevant rules and formulas. Matrix multiplication is associative, which means $A(BC)=(AB)C$ for conformable sizes. Matrix multiplication is distributive over addition, which means $A(B+C)=AB+AC$ and $(B+C)A=BA+CA$ when sizes match. If $I$ is the appropriate identity matrix then $AI=A$ and $IA=A$. Matrix multiplication is not generally commutative; in general $AB\neq BA$. 3. Counterexample showing which property fails. Choose $$A=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}$$. Compute $$AB=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}$$. Compute $$BA=\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}=\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}$$. Since $$AB\neq BA$$, commutativity (option D) fails in general. 4. Final answer. The property that does not hold for matrix multiplication is D. $BA=AB$.