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Figure 1.3.1
Transpose of a Matrix
We conclude this section by defining two matrix operations that have no analogs in the real numbers.
DEFINITION
If A is any matrix, then the transpose of A, denoted by , is defined to be the matrix that results from
interchanging the rows and columns of A; that is, the first column of is the first row of A, the second column of is
the second row of A, and so forth.
EXAMPLE 11 Some Transposes
The following are some examples of matrices and their transposes.
Observe that not only are the columns of the rows of A, but the rows of are the columns of A. Thus the entry in row i
