Vectors and Matrices

Matrix Transpose

Transpose a 2x3 matrix by placing each entry A[i,j] at position A^T[j,i] in the result. Each entry is relocated in one step; the state panel accumulates the transposed matrix one cell at a time. The result is a 3x2 matrix.

transpose The transpose A^T of an m x n matrix A is an n x m matrix defined by A^T[j,i] = A[i,j]. Rows of A become columns of A^T.

Example

Move each matrix entry into its transposed position one entry at a time.

highlighted = computed this step

Step 1 — Set up

Start with the matrix and an empty transpose.

A=[123456],AT=[□□□□□□]A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix},\quad A^T=\begin{bmatrix}\square&\square\\\square&\square\\\square&\square\end{bmatrix}

Step 2 — Top-Left entry

Move the top-left entry to the top-left slot: 1.

1→1,AT=[1□□□□□]1\rightarrow 1,\quad A^T=\begin{bmatrix}\hl{1}&\square\\\square&\square\\\square&\square\end{bmatrix}

Step 3 — Top-Middle entry

Move the top-middle entry to the middle-left slot: 2.

2→2,AT=[1□2□□□]2\rightarrow 2,\quad A^T=\begin{bmatrix}1&\square\\\hl{2}&\square\\\square&\square\end{bmatrix}

Step 4 — Top-Right entry

Move the top-right entry to the bottom-left slot: 3.

3→3,AT=[1□2□3□]3\rightarrow 3,\quad A^T=\begin{bmatrix}1&\square\\2&\square\\\hl{3}&\square\end{bmatrix}

Step 5 — Bottom-Left entry

Move the bottom-left entry to the top-right slot: 4.

4→4,AT=[142□3□]4\rightarrow 4,\quad A^T=\begin{bmatrix}1&\hl{4}\\2&\square\\3&\square\end{bmatrix}

Step 6 — Bottom-Middle entry

Move the bottom-middle entry to the middle-right slot: 5.

5→5,AT=[14253□]5\rightarrow 5,\quad A^T=\begin{bmatrix}1&4\\2&\hl{5}\\3&\square\end{bmatrix}

Step 7 — Bottom-Right entry

Move the bottom-right entry to the bottom-right slot: 6.

6→6,AT=[142536]6\rightarrow 6,\quad A^T=\begin{bmatrix}1&4\\2&5\\3&\hl{6}\end{bmatrix}

Step 8 — Result

Read the completed transpose.

AT=[142536]A^T=\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}