# Matrix Transposes

The transpose operation is one that allows us to swap the rows and columns of a matrix. For example, suppose we want to transpose $\left[\begin{array}{cc} 4 & 5 & 7 \\ 2 & 3 & 4 \end{array}\right]$. We would change the order of the row and column index, giving us $\left[\begin{array}{cc} 4 & 2 \\ 5 & 3 \\ 7 & 4 \end{array}\right]$

Formally, we define the transpose operator as follows. If you have a matrix A, the transpose, $A^t$, is defined as: $(A^T)_{ij} = (A)_{ji}$

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:

- $(A^t)^T = A$
- $(A+B)^T = A^T + B^T$
- $(sA)^T = sA^T$