Transpose
Products

If A is an m × n matrix and A is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A is m × m and A A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A has entries that are the inner product of a row of A with a column of A. But the columns of A are the rows of A, so the entry corresponds to the inner product of two rows of A. If p is the entry of the product, it is obtained from rows i and j in A. The entry p is also obtained from these rows, thus p = p, and the product matrix (p) is symmetric. Similarly, the product A A is a symmetric matrix.

A quick proof of the symmetry of A A results from the fact that it is its own transpose:
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