If A is 5×75 imes7 then A⊤A^ op is [Fill in the blank],
× imes [Fill in the blank],
.

Question

If A is 5×75	imes7 then A⊤A^	op is [Fill in the blank],
 ×	imes  [Fill in the blank],
.

BlackTom AI · Accepted Answer

Carefully restating the prompt and what is being asked can help, since the original text has some typographical ambiguity. The question as given says: If A is 5×7, then A⊤A^⊤ is [Fill in the blank], × [Fill in the blank]. The provided answer values (according to the supplied data) are 7 and 5 for the two blanks.

Option-by-option analysis:
- What the blanks are attempting to capture: typically, when you see a statement like A is 5×7, the dimensions of A^T (the transpose) are 7×5. For a product with a transpose, common products are A A^T (which would be 5×5) or A^T A (which would be 7×7). The phrasing A^T A^T is unusual because it would require multiplying a 7×5 matrix by another 7×5 matrix, which is not defined (the inner dimensions would be 5 and 7, not matching), so that product is not defined.

- If we take the given blanks as literally asking for the dimensions of A^T or of A^T times something, we should compare with the standard results:
  - A is 5×7, so A^T is 7×5.
  - The product A A^T would be 5×5.
  - The product A^T A would be 7×7.
  - The expression A^T A^T, as written, would not be a valid product for A of size 5×7, since you’d be trying to multiply a 7×5 matrix by a 7×5 matrix, which is undefined due to inner dimensions not matching (5 ≠ 7).

- Given the explicit answer blanks are 7 and 5, we can interpret how those numbers could appear:
  - One potential interpretation is that the intended blanked form was simply to state the shape of A^T, which is 7×5. In that reading, the first blank would be 7 (the number of rows of A^T), and the second blank would be 5 (the number of columns of A^T).
  - Another reading is that the problem intended a product like A^T A or A A^T, but the blanks do not align with the resulting dimensions of those products (7×7 or 5×5, respectively). If someone wrote A^T A^T by mistake, the fixed dimensions 7 and 5 do not resolve into a valid product, so the item would be flawed as stated.

- Why the option of 7 and 5 could be tempting:
  - It matches the dimensions of A^T itself (7×5). If the question was simply asking for the dimensions of A^T, the blanks would indeed be 7 and 5.
  - It’s easy to misread a prompt with a transpose and end up thinking about A^T’s shape rather than a valid matrix product.

- Why this makes the other interpretations incorrect in typical practice:
  - If the intended product were A A^T, the result would be 5×5, not 7×5, so the pair (7, 5) would not describe that product.
  - If the intended product were A^T A, the result would be 7×7, not 7×5, so again the pair (7, 5) would be incorrect for that product.

- In summary, with A defined as 5×7, the standard, well-defined products give either 5×5 or 7×7 results. The specific combination A^⊤A^⊤ is not a valid product as written. The blanks 7 and 5 best align with describing the shape of A^T itself (7×5) rather than a valid product of A^T with itself, so the prompt appears to be inconsistent if interpreted strictly as a matrix product query.

MAT223H1S Winter 2025 Reading Quiz 4

If A is 5×75\times7 then A⊤A^\top is [Fill in the blank], ×\times [Fill in the blank], .

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