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Here, the dimension of each transpose is identical with that of the original matrix.

In D', we also note the remarkable result that D' inherits not only the dimension of D but also the original array of elements! The fact that D' = D is the result of the symmetry of the elements with reference to the principal diagonal. Considering the principal diagonal in D as a mirror, the elements "

located to its northeast are exact images of the elements to its southwest; hence the first row reads identically with the first column, and so forth. The matrix D exemplifies the special class of square matrices known as symmetric matrices. Another example of such a matrix is the identity matrix I, which, as a symmetric matrix, has the transpose /' = I.

Properties of Transposes

The following properties characterize transposes:

The first says that the transpose of the transpose is the original matrix—a rather self-evident conclusion.

The second property may be verbally stated thus: the transpose of a sum is the sum of the transposes.

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