The transpose operation is one that allows us to swap the rows and columns of a matrix. For example, suppose we want to transpose . We would change the order of the row and column index, giving us

Formally, we define the transpose operator as follows. If you have a matrix A, the transpose, , is defined as:

There are a few properties of transposing matrices that are helpful to know. If A and B are matrices and s is a real number scalar, then:

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