Let's analyze each statement one by one, providing reasoning or counterexamples as needed:

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a. If rows 1 and 3 of 𝐵 are the same, so are rows 1 and 3 of 𝐴𝐵.

• True. When multiplying two matrices, the rows of the resulting matrix $A \times B$ are linear combinations of the rows of $A$, with the coefficients being the elements of $B$. If rows 1 and 3 of $B$ are the same, then the linear combinations (i.e., the rows) in the product matrix $A \times B$ will also be the same because the coefficients for rows 1 and 3 in $B$ are identical. Therefore, rows 1 and 3 of $A \times B$ will also be the same.

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b. If columns 1 and 3 of 𝐵 are the same, so are the columns 1 and 3 of 𝐴𝐵.

• False. While the columns of the resulting product matrix $A \times B$ are linear combinations of the columns of $B$ (weighted by the corresponding rows of $A$), the fact that columns 1 and 3 of $B$ are the same doesn't necessarily imply that columns 1 and 3 of $A \times B$ will be the same. This is because the rows of $A$ can affect the way the columns of $B$ are combined. A counterexample: if $A$ has non-identical rows, the linear combinations might not preserve the equality between columns 1 and 3 of $A \times B$.

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c. If rows 1 and 3 of 𝐴 are the same, so are rows 1 and 3 of 𝐴𝐵𝐶.

• True. In the product $A \times B \times C$, the rows of $A$ are used as the basis for the rows of the resulting product. If rows 1 and 3 of $A$ are the same, then for any matrix multiplication afterward (with compatible matrices), these rows will remain the same in the final product. Thus, rows 1 and 3 of $A \times B \times C$ will also be the same.

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d. If 𝐴𝐵 and 𝐵𝐴 are both defined, then 𝐴𝐵 and 𝐵𝐴 are both square (though possibly of different sizes).

• False. If $A \times B$ and $B \times A$ are both defined, the number of rows and columns of $A$ and $B$ must align appropriately. However, this does not imply that the resulting matrices must be square. The product $A \times B$ could result in a rectangular matrix, and the product $B \times A$ could be of a different rectangular shape. A counterexample: Let $A$ be a $2 \times 3$ matrix and $B$ be a $3 \times 2$ matrix. Both $A \times B$ (which results in a $2 \times 2$ matrix) and $B \times A$ (which results in a $3 \times 3$ matrix) are defined, but neither is square.

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e. $(A \times B)^2 = B^2 \times A^2.$

• False. In general, matrix multiplication is not commutative, meaning that $A \times B \neq B \times A$ in most cases. Therefore, squaring $A \times B$ will not give the same result as squaring $B \times A$. In fact, $(A \times B)^2 = A \times B \times A \times B$, which is not the same as $B^2 \times A^2$ unless special conditions hold (like commutative matrices, which is rare in practice).

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Summary of answers:

• True statements: a, c
• False statements: b, d, e

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Would you like more detailed explanations or have any follow-up questions?

Here are five related questions for further exploration:

1. What happens when both rows and columns of two matrices are identical?
2. How does matrix multiplication affect the symmetry of matrices?
3. Can two non-square matrices yield a square product?
4. Under what conditions can the matrix multiplication $(A \times B)^2 = B^2 \times A^2$ hold?
5. How does the rank of a matrix change after multiplying with another matrix?

Tip: Always verify matrix dimensions before performing matrix multiplication to ensure compatibility and avoid errors in calculations!