Title. The construction where a matrix A is, basically, squared by multiplying by its transpose is commonly used, but the construction obscures the similarity to squaring for single numbers.
rule 1: specifically wondering about the connection between algebra and linear algebra/algebra for matrices. to what extent does it make sense to extend algebra to operations on sets of numbers?
Cause it would be a huge pain in the ass. For example then A^(-1)A² wouldn't be A. Also what would A³ be? General integer powers of matrices are common.
This question makes no sense to me. The transpose has a specific definition, and so does matrix multiplication. In general, AA^(T) ≠ A^(2). What does it even mean to "define" AA^(T) as A^(2)?
Because AA^T and A^2 aren't always equal to each other.

In fact, A^2 may not even be defined when AA^T is.

If A is a 2x3 matrix, then AA^T will be a 2x2 matrix, but A^2 can't be computed, because you can't multiply a 2x3 matrix by itself.
Because they are different. Not sure what you mean exactly
Let A be a MxN matrix. B=A^T A is a MxM matrix while C=A A^T is a NxN matrix.

A^2 is commutative so A^2 = A^T A = A A^T, which means B=C.