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# Matrix Times Its Transpose

The product of a matrix and its transpose is known as the matrix squared. It is denoted as A^2.

The matrix squared is a special case of the matrix product, which is defined as follows:

If A is an m x n matrix and B is an n x p matrix, then the matrix product of A and B is the m x p matrix C, where each element c_ij is given by:

c_ij = ∑_(k=1)^n▒a_ik * b_kj

The transpose of a matrix A is denoted as A^T and is obtained by reflecting A over its main diagonal (which runs from top left to bottom right). The element at row i and column j of A becomes the element at row j and column i of A^T.

For example, if A is a 3 x 2 matrix:

[a_11, a_12] [a_21, a_22] [a_31, a_32]

Then the transpose of A, A^T, is a 2 x 3 matrix:

[a_11, a_21, a_31] [a_12, a_22, a_32]

The matrix squared, A^2, is defined as the product of A and its transpose, A^T. So, if A is a 3 x 2 matrix, then A^2 is a 3 x 3 matrix:

[a_11^2 + a_12^2, a_11a_21 + a_12a_22, a_11a_31 + a_12a_32] [a_21a_11 + a_22a_12, a_21^2 + a_22^2, a_21a_31 + a_22a_32] [a_31a_11 + a_32a_12, a_31a_21 + a_32a_22, a_31^2 + a_32^2]