Let's analyze each of the statements carefully.

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

• True: For 𝐴𝐵 to be defined, the number of columns of 𝐴 must match the number of rows of 𝐵. Similarly, for 𝐵𝐴 to be defined, the number of columns of 𝐵 must match the number of rows of 𝐴. If both are defined, this suggests that the dimensions of 𝐴 and 𝐵 are compatible in such a way that both 𝐴𝐵 and 𝐵𝐴 are square matrices. However, the sizes may be different; for example, 𝐴 could be $2 \times 3$ and 𝐵 could be $3 \times 2$, making both products $2 \times 2$, so both could be square but of different sizes.

b. (𝐴𝐵)² = 𝐵²𝐴².

• False: Matrix multiplication is not commutative, so in general, the equality $(𝐴𝐵)² = 𝐵²𝐴²$ does not hold. The left-hand side involves $(𝐴𝐵)(𝐴𝐵)$, while the right-hand side involves $𝐵𝐵𝐴𝐴$. Since the order of matrix multiplication matters, this equality is not valid in general.

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

• True: If rows 1 and 3 of 𝐴 are the same, then multiplying 𝐴 by any matrix will preserve that equality. This is because the multiplication of 𝐴 with 𝐵 and 𝐶 does not affect the identical nature of rows 1 and 3 of 𝐴. This can be confirmed by testing with specific examples of matrices.

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

• False: While multiplying by 𝐴 does not necessarily change the rows of 𝐵, it can alter the structure of the product. Specifically, row 1 and row 3 of 𝐵 being the same does not guarantee that the resulting rows in 𝐴𝐵 will be the same, because multiplication by 𝐴 can combine columns in different ways that affect the row structure.

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

• True: If columns 1 and 3 of 𝐵 are the same, then multiplying by 𝐴 does not alter the equality of these columns. This can be checked by observing how multiplying a matrix 𝐵 with identical columns by another matrix 𝐴 preserves the equality in the resulting product.

Final Answer:

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

Would you like further clarification on any of the statements, or do you have another question?

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Additional Questions:

1. Why does matrix multiplication fail to be commutative?
2. What happens when you multiply a matrix by an identity matrix?
3. Can the order of matrix multiplication ever be rearranged without changing the result?
4. How can you test if two matrices are invertible?
5. What role does the concept of a "square matrix" play in matrix operations?

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Tip: Always check the dimension compatibility first when analyzing matrix products.