The TransposeSteps command is used to show the steps of finding the transpose of a matrix.

The displaystyle and output options can be used to change the output format.  See OutputStepsRecord for details.

This function is part of the Student[LinearAlgebra] package, so it can be used in the short form TransposeSteps(..) only after executing the command with(Student[LinearAlgebra]). However, it can always be accessed through the long form of the command by using Student[LinearAlgebra][TransposeSteps](..).

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$P≔\mathrm{Matrix}\left(\left[\left[1\,3\,2\right]\,\left[2\,3\,1\right]\,\left[2\,2\,1\right]\right]\right)$

$P≔\left[\begin{array}{ccc}1& 3& 2\\ 2& 3& 1\\ 2& 2& 1\end{array}\right]$

$\mathrm{TransposeSteps}\left(P\right)$

$\begin{array}{lll}& & M=\left[\begin{array}{ccc}1& 3& 2\\ 2& 3& 1\\ 2& 2& 1\end{array}\right]\\ \text{•}& & \text{Recall the definition of transpose. Given:}\\ & & A=\left[\begin{array}{ccc}''A1''& ''A2''& ''A3''\\ ''B1''& ''B2''& ''B3''\\ ''C1''& ''C2''& ''C3''\end{array}\right]\\ \text{•}& & \text{The rows are switched to columns.}\\ & & A^T=\left[\begin{array}{ccc}''A1''& ''B1''& ''C1''\\ ''A2''& ''B2''& ''C2''\\ ''A3''& ''B3''& ''C3''\end{array}\right]\\ \text{•}& & \text{Take the first row of the given matrix.}\\ & & \left[\begin{array}{ccc}1& 3& 2\\ \mathrm{..}& \mathrm{..}& \mathrm{..}\\ \mathrm{..}& \mathrm{..}& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Insert it as the first column of the result matrix.}\\ & & \left[\begin{array}{ccc}1& \mathrm{..}& \mathrm{..}\\ 3& \mathrm{..}& \mathrm{..}\\ 2& \mathrm{..}& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Follow the pattern to copy all rows to columns. The transpose is:}\\ & & M^T=\left[\begin{array}{ccc}1& 2& 2\\ 3& 3& 2\\ 2& 1& 1\end{array}\right]\end{array}$

$A≔⟨⟨8\,4\,-1\,-5⟩\left|⟨4\,2\,-0.5\,-2.5⟩\right|⟨-5\,-2\,3\,-1⟩|⟨-5\,-2.5\,-4\,-9⟩⟩$

$A≔\left[\begin{array}{cccc}8& 4& \mathrm{-5}& \mathrm{-5}\\ 4& 2& \mathrm{-2}& \mathrm{-2.5}\\ \mathrm{-1}& \mathrm{-0.5}& 3& \mathrm{-4}\\ \mathrm{-5}& \mathrm{-2.5}& \mathrm{-1}& \mathrm{-9}\end{array}\right]$

$\mathrm{TransposeSteps}\left(A\right)$

$\begin{array}{lll}& & M=\left[\begin{array}{cccc}8& 4& \mathrm{-5}& \mathrm{-5}\\ 4& 2& \mathrm{-2}& \mathrm{-2.5}\\ \mathrm{-1}& \mathrm{-0.5}& 3& \mathrm{-4}\\ \mathrm{-5}& \mathrm{-2.5}& \mathrm{-1}& \mathrm{-9}\end{array}\right]\\ \text{•}& & \text{Recall the definition of transpose. Given:}\\ & & A=\left[\begin{array}{cccc}''A1''& ''A2''& ''A3''& ''A4''\\ ''B1''& ''B2''& ''B3''& ''B4''\\ ''C1''& ''C2''& ''C3''& ''C4''\\ ''D1''& ''D2''& ''D3''& ''D4''\end{array}\right]\\ \text{•}& & \text{The rows are switched to columns.}\\ & & A^T=\left[\begin{array}{cccc}''A1''& ''B1''& ''C1''& ''D1''\\ ''A2''& ''B2''& ''C2''& ''D2''\\ ''A3''& ''B3''& ''C3''& ''D3''\\ ''A4''& ''B4''& ''C4''& ''D4''\end{array}\right]\\ \text{•}& & \text{Take the first row of the given matrix.}\\ & & \left[\begin{array}{cccc}8& 4& \mathrm{-5}& \mathrm{-5}\\ \mathrm{..}& \mathrm{..}& \mathrm{..}& \mathrm{..}\\ \mathrm{..}& \mathrm{..}& \mathrm{..}& \mathrm{..}\\ \mathrm{..}& \mathrm{..}& \mathrm{..}& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Insert it as the first column of the result matrix.}\\ & & \left[\begin{array}{cccc}8& \mathrm{..}& \mathrm{..}& \mathrm{..}\\ 4& \mathrm{..}& \mathrm{..}& \mathrm{..}\\ \mathrm{-5}& \mathrm{..}& \mathrm{..}& \mathrm{..}\\ \mathrm{-5}& \mathrm{..}& \mathrm{..}& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Follow the pattern to copy all rows to columns. The transpose is:}\\ & & M^T=\left[\begin{array}{cccc}8& 4& \mathrm{-1}& \mathrm{-5}\\ 4& 2& \mathrm{-0.5}& \mathrm{-2.5}\\ \mathrm{-5}& \mathrm{-2}& 3& \mathrm{-1}\\ \mathrm{-5}& \mathrm{-2.5}& \mathrm{-4}& \mathrm{-9}\end{array}\right]\end{array}$

$\mathrm{TransposeSteps}\left(\mathrm{Matrix}\left(\left[\left[7\,5\right]\,\left[2\,3\right]\right]\right)\,\mathrm{output}=\mathrm{printf}\right)$

        M = (Matrix(2, 2, [[7,5],[2,3]]))
• Recall the definition of transpose.
Given:
        A = (Matrix(2, 2, [[A1,A2],[B1,B2]]))
• The rows are switched to columns.
        A^T = (Matrix(2, 2, [[A1,B1],[A2,B2]]))
• Take the first row of the given matrix.
        Matrix(2, 2, [[7,5],[..,..]])
• Insert it as the first column of the result matrix.
        Matrix(2, 2, [[7,..],[5,..]])
• Follow the pattern to copy all rows to columns.  The transpose is:
        M^T = (Matrix(2, 2, [[7,2],[5,3]]))

$\mathrm{TransposeSteps}\left(\mathrm{Matrix}\left(\left[\left[7\,5\,11\right]\,\left[2\,13\,8\right]\right]\right)\right)$

$\begin{array}{lll}& & M=\left[\begin{array}{ccc}7& 5& 11\\ 2& 13& 8\end{array}\right]\\ \text{•}& & \text{Recall the definition of transpose. Given:}\\ & & A=\left[\begin{array}{ccc}''A1''& ''A2''& ''A3''\\ ''B1''& ''B2''& ''B3''\end{array}\right]\\ \text{•}& & \text{The rows are switched to columns.}\\ & & A^T=\left[\begin{array}{cc}''A1''& ''B1''\\ ''A2''& ''B2''\\ ''A3''& ''B3''\end{array}\right]\\ \text{•}& & \text{Take the first row of the given matrix.}\\ & & \left[\begin{array}{ccc}7& 5& 11\\ \mathrm{..}& \mathrm{..}& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Insert it as the first column of the result matrix.}\\ & & \left[\begin{array}{cc}7& \mathrm{..}\\ 5& \mathrm{..}\\ 11& \mathrm{..}\end{array}\right]\\ \text{•}& & \text{Follow the pattern to copy all rows to columns. The transpose is:}\\ & & M^T=\left[\begin{array}{cc}7& 2\\ 5& 13\\ 11& 8\end{array}\right]\end{array}$

$\,\mathrm{mi}\left(''TransposeSteps''\right)\,\mathrm{mo}\left(\right)\,\mathrm{mfenced}\left(\mathrm{mi}\left(\mathrm{mi}\left(\left(\right)\left(''Vector''\,\mathrm{fontstyle}\=''normal''\right)\,\left(\mathrm{fontstyle}\=''normal''\right)\left(\left(\right)\left(''row''\right)\,\mathrm{open}\=''[''\,\mathrm{close}\='']''\right)\right)\,\,\left(\mathrm{fontstyle}\=''normal''\right)\left(\left(\mathrm{fontstyle}\=''normal''\right)\left(\mathrm{mn}\left(''1''\right)\,\left(\right)\left(''2''\right)\,\,''TransposeSteps''\=\mathrm{mo}\right)\right)\right)\right)$

$\begin{array}{lll}& & V=\left[\begin{array}{cc}1& 2\end{array}\right]\\ \text{•}& & \text{The transpose of a row vector is a column vector with the same entries:}\\ & & V^T=\left[\begin{array}{c}1\\ 2\end{array}\right]\end{array}$

$\mathrm{TransposeSteps}\left(\mathrm{Vector}\left[\mathrm{column}\right]\left(\left[1\,2\right]\right)\right)$

$\begin{array}{lll}& & V=\left[\begin{array}{c}1\\ 2\end{array}\right]\\ \text{•}& & \text{The transpose of a column vector is a row vector with the same entries:}\\ & & V^T=\left[\begin{array}{cc}1& 2\end{array}\right]\end{array}$

The Student[LinearAlgebra][TransposeSteps] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.