Transpose

The transpose of a matrix A is another matrix where the rows of A are written as columns. Vectors can be transposed in the same way. We can write the transpose of A using different symbols such as A^T, A′ A^tr and A^t.

Examples

Here is the vector $\left[\begin{array}{cc}1& 2\end{array}\right]$ being transposed:

• ${\left[\begin{array}{cc}1& 2\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{c}1\\ 2\end{array}\right].$

Here are a few matrices being transposed:

• ${\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right].$

• ${\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right].$

• ${\left[\begin{array}{ccc}1& 2& 8\\ 3& 4& 3\\ 5& 6& 1\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\\ 8& 3& 1\end{array}\right].$

Properties

Given two matrices A and B, the following properties related to the transpose are true:

• $(A^T{)}^{-1}=(A^{-1}{)}^T$
• $(AB{)}^T=B^T{A}^T$

Related pages

• Invertible matrix