We are tasked with showing that for two matrices $A$ and $B$, in general, $A \times B \neq B \times A$ unless $A = B$. This is a property of non-commutative multiplication for matrices.

General Idea:

In matrix multiplication, the order in which matrices are multiplied matters, meaning that the product of $A \times B$ is generally not the same as $B \times A$. This is because the entries of the resulting matrices depend on the specific rows and columns involved in the multiplication, which differ when the order is swapped.

Proof by Example:

Let’s illustrate this with an example to show that $A \times B \neq B \times A$ unless $A = B$.

Consider two matrices $A$ and $B$ as follows:

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$

Step 1: Compute $A \times B$

$A \times B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$

= \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$$ #### Step 2: Compute $$B \times A$$ $$B \times A = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ $$= \begin{pmatrix} (5 \cdot 1 + 6 \cdot 3) & (5 \cdot 2 + 6 \cdot 4) \\ (7 \cdot 1 + 8 \cdot 3) & (7 \cdot 2 + 8 \cdot 4) \end{pmatrix} = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}$$ #### Step 3: Compare $$A \times B$$ and $$B \times A$$ We can see that: $$A \times B = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}, \quad B \times A = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}$$ Clearly, $$A \times B \neq B \times A$$. Hence, the product of matrices is generally not commutative. ### Special Case: $$A = B$$ Now, if $$A = B$$, then clearly: $$A \times B = B \times A = A \times A$$ Thus, in the specific case where $$A = B$$, the products are equal. ### Conclusion: In general, $$A \times B \neq B \times A$$ for matrices unless