To solve the problem step by step, we will prove that $(AB)^T = B^T A^T$ given the matrices:

$A = \begin{bmatrix} 1 & 3 \\ 1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}.$

Step 1: Matrix Multiplication ($AB$)

First, compute the product $AB$: $AB = \begin{bmatrix} 1 & 3 \\ 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}.$

The resulting matrix is calculated as: $AB = \begin{bmatrix} (1 \cdot 1 + 3 \cdot 1) & (1 \cdot 2 + 3 \cdot 1) \\ (1 \cdot 1 + 2 \cdot 1) & (1 \cdot 2 + 2 \cdot 1) \end{bmatrix}.$

Performing the calculations: $AB = \begin{bmatrix} 4 & 5 \\ 3 & 4 \end{bmatrix}.$

Step 2: Transpose of $AB$ ($(AB)^T$)

Now take the transpose of the resulting matrix $AB$: $(AB)^T = \begin{bmatrix} 4 & 5 \\ 3 & 4 \end{bmatrix}^T.$

Transpose is achieved by swapping rows and columns: $(AB)^T = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}.$

Step 3: Transposes of $B$ and $A$

Compute the transpose of $B$ ($B^T$) and $A$ ($A^T$).

For $B^T$: $B = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}, \quad B^T = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}.$

For $A^T$: $A = \begin{bmatrix} 1 & 3 \\ 1 & 2 \end{bmatrix}, \quad A^T = \begin{bmatrix} 1 & 1 \\ 3 & 2 \end{bmatrix}.$

Step 4: Compute $B^T A^T$

Now compute the product $B^T A^T$: $B^T A^T = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 3 & 2 \end{bmatrix}.$

The resulting matrix is calculated as: $B^T A^T = \begin{bmatrix} (1 \cdot 1 + 1 \cdot 3) & (1 \cdot 1 + 1 \cdot 2) \\ (2 \cdot 1 + 1 \cdot 3) & (2 \cdot 1 + 1 \cdot 2) \end{bmatrix}.$

Performing the calculations: $B^T A^T = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}.$

Step 5: Verify $(AB)^T = B^T A^T$

We already computed: $(AB)^T = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}, \quad B^T A^T = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}.$

Since $(AB)^T = B^T A^T$, the equality is proven.

Final Answer:

$(AB)^T = B^T A^T$

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Here are 5 related questions for further exploration:

1. What is the general formula for the transpose of a product of matrices?
2. How does associativity in matrix multiplication relate to this proof?
3. What are the conditions for two matrices to be conformable for multiplication?
4. How would the proof change if the matrices were not square?
5. Can this property be extended to three or more matrices?

Tip: Always verify the dimensions of matrices before performing operations like multiplication or transposition!