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If $B = A^T$ where $B$ is an $m \times n$ matrix, find the size of $A, AA^2, A^{T}A$.

We know that since $B = A^T$, and $B = m \times n$ matrix, that means $A$ is $n \times m$ matrix.

What about $AA^2$?

That is, $A (A \times A) = (n \times m)((n \times m)\times (n \times m))$

For this multiplication to be defined, we must have $n=m$, in order words, square matrix. My question is though, in the back it says the answer is $n \times n$ matrix? Can I also say that it is $m \times m$? Is there more then one correct answer in this case?

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    $A^2$ is not defined when it is not a square matrix. – 2017-01-22
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    I know, but my question is, would I be valid in saying the $AA^2$ is a $m \times m$? – 2017-01-22
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    You would not be. – 2017-01-22
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    When $n=m$, what is the difference between $n\times n$ and $m\times m$? – 2017-01-22
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    @CameronWilliams why not? and Jack I ask because my textbook says a specific one, instead of "any", so I was just curious – 2017-01-22
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    Judging by the rest of the question, my guess is that "$AA^2$" is a typo and that it's supposed to be "$AA^T$". – 2017-01-22
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    Its not a typo. – 2017-01-22

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$A^2$ is not defined when it is not a square matrix.

The question regarding the size of $A^2$ is meaningful only when $A$ is a square matrix, i.e., it is of the size $n\times n$ for some positive integer $n$.

You have said in your question that "For this multiplication to be defined, we must have $n=m$". In this case, there is no difference between $n\times n$ and $m\times m$.