Algebra as Structure Math Example 2

Solution

1. 1
   Step 1: Numbers: $3 \times 4 = 4 \times 3 = 12$. Commutative ✓
2. 2
   Step 2: Matrices: $AB \neq BA$ in general.
3. 3
   Step 3: Example: $\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}0&1\\0&0\end{bmatrix} = \begin{bmatrix}0&1\\0&0\end{bmatrix}$ but reversed $= \begin{bmatrix}0&0\\0&0\end{bmatrix}$.
4. 4
   Different structure! Matrix multiplication is associative but NOT commutative.