TeachingLinear Algebra – Matrices – The transpose of a matrix
Definition: If \(A=(a_{ij})\) is an \(m\times n\) matrix, then \(A^T\) is the \(n\times m\) matrix given by \(A^T=(a_{ji})\).
Theorem: If \(A\) is an \(m\times n\) matrix and \(B\) is an \(n\times p\) matrix, then: \((AB)^T=B^TA^T\).
This important property can be used as follows.
Example: Let \(A=\begin{pmatrix}1&2\\-1&3\end{pmatrix}\) and \(B=\begin{pmatrix}3&-4\\2&-1\end{pmatrix}\). Then the problem of finding a matrix \(X\) such that \(AX=B\) is easily solved using row reduction:
\[\left(\;\left.\begin{matrix}1&2\\-1&3\end{matrix}\;\right|\;\begin{matrix}3&-4\\2&-1\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&2\\0&5\end{matrix}\;\right|\;\begin{matrix}3&-4\\5&-5\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&2\\0&1\end{matrix}\;\right|\;\begin{matrix}3&-4\\1&-1\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&0\\0&1\end{matrix}\;\right|\;\begin{matrix}1&-2\\1&-1\end{matrix}\;\right) \quad\Longrightarrow\quad X=\begin{pmatrix}1&-2\\1&-1\end{pmatrix}.\]However, how to find a matrix \(Y\) such that \(YA=B\)? Of course, one can set \(Y=\begin{pmatrix}y_1&y_2\\y_3&y_4\end{pmatrix}\) and obtain a system of four linear equations for the four unknowns. Instead, it is much more elegant to use the transpose:
\[YA=B\quad\Longleftrightarrow\quad(YA)^T=B^T\quad\Longleftrightarrow\quad A^TY^T=B^T.\]This can be solved as above using row reduction:
\[\left(\;\left.\begin{matrix}1&-1\\2&3\end{matrix}\;\right|\;\begin{matrix}3&2\\-4&-1\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&-1\\0&5\end{matrix}\;\right|\;\begin{matrix}3&2\\-10&-5\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&-1\\0&1\end{matrix}\;\right|\;\begin{matrix}3&2\\-2&-1\end{matrix}\;\right)\sim \left(\;\left.\begin{matrix}1&0\\0&1\end{matrix}\;\right|\;\begin{matrix}1&1\\-2&-1\end{matrix}\;\right) \quad\Longrightarrow\quad Y^T=\begin{pmatrix}1&1\\-2&-1\end{pmatrix}.\]This implies that \(Y=\begin{pmatrix}1&-2\\1&-1\end{pmatrix}\).
Last modified on March 1, 2021
