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You can transpose any matrix, regardless of how many rows and columns it has. Square matrices, with an equal number of rows and columns, are most commonly transposed, so we'll use a simple square matrix as an example: matrix A =1  2  34  5  67  8  9 Rewrite row one of the matrix as a column:  transpose of matrix A = AT  first column of AT:123 The second row of the original matrix becomes the second column of its transpose. Repeat this pattern until you have turned every row into a column:  AT =1  4  72  5  83  6  9 The transposition is exactly the same for a non-square matrix. You rewrite the first row as the first column, the second row as the second column, and so forth. Here's an example with color-coding to show you where the elements end up:  matrix Z =4  7  2  13  9  8  6  matrix ZT =4  37  92  81  6 The concept is pretty simple, but it's good to be able to describe it in mathematics. No jargon is required beyond basic matrix notation:  If matrix B is an m x n matrix (m rows and n columns), the transposed matrix BT is an n x m matrix (n rows and m columns).  For each element bxy (xth row, yth column) in B, the matrix BT has an equal element at byx (yth row, xth column).

Summary:
Start with any matrix. Turn the first row of the matrix into the first column of its transpose. Repeat for the remaining rows. Practice on a non-square matrix. Express the transposition mathematically.