Final answer:
Multiplying a matrix by its transpose results in a symmetric matrix for square matrices, with each element equal to the dot product of the corresponding row and column vectors.
Step-by-step explanation:
When you multiply a matrix by its transpose, several results are possible, depending on the properties of the original matrix. For a square matrix, the product is a symmetric matrix. This product is not necessarily commutative, meaning AB is not always equal to BA, where A is the original matrix, and B is its transpose. In general, the resulting matrix will have each element (i,j) equal to the dot product of the i-th row of the original matrix and the j-th column of the transpose. The diagonal elements of the resulting matrix represent the scalar product of vectors where the corresponding row and column are identical, showing the sum of squares of each entry in that row or column of the original matrix.