AcademyMatrix Algebra
Academy
Transpose
Level 1 - Linear Algebra topic page in Matrix Algebra.
Principle
The transpose of a matrix swaps rows and columns. Row \(i\) becomes column \(i\), and column \(j\) becomes row \(j\). Transpose is a way to reverse the orientation of matrix data.
Notation
\(A^T\)
the transpose of matrix A
\(a_{ij}\)
entry of A in row i and column j
\((A^T)_{ij}\)
entry of the transpose in row i and column j
\(\mathbf r_i\)
row i of A
\(\mathbf c_i\)
column i of A
The Core Method
If \(A\) is \(m\) by \(n\), then \(A^T\) is \(n\) by \(m\), and
Transpose entries
\[(A^T)_{ij}=a_{ji}\]
Transpose obeys these rules whenever the expressions are defined:
Double transpose
\[(A^T)^T=A\]
Transpose of a sum
\[(A+B)^T=A^T+B^T\]
Transpose of a product
\[(AB)^T=B^TA^T\]
The product rule reverses the order. This reversal is essential.
Worked Cases
Question
Find the transpose of \(A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\).
Answer
The rows of \(A\) become columns of \(A^T\). Row \(1\), \((1,2,3)\), becomes column \(1\). Row \(2\), \((4,5,6)\), becomes column \(2\). Therefore \(A^T=\begin{pmatrix}1&4\\2&5\\3&6\end{pmatrix}\).
Examples
Question
Transpose \(\begin{pmatrix}2\\-1\\5\end{pmatrix}\).
Answer
A \(3\) by \(1\) column matrix transposes to a \(1\) by \(3\) row matrix: \(\begin{pmatrix}2\\-1\\5\end{pmatrix}^T=\begin{pmatrix}2&-1&5\end{pmatrix}\).