## Info

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 matrixâ€”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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