Transposing matrices

Transposing a matrix just means we make the rows into columns and the columns into rows. Or, in other words, we somewhat flip the matrix across its diagonal.

For example, if we have the matrix:

\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

The transpose of \mathbf{A}, which we write as \mathbf{A}^T, is:

\mathbf{A}^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}

That’s because the first row, [1 \quad 2 \quad 3], becomes the first column, and the second row, [4 \quad 5 \quad 6], becomes the second column.

Example transpositions

Find the transpose of \mathbf{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}

• The first row is [7 \quad 8], which becomes the first column, so the matrix will look like this so far:
  • \begin{bmatrix} 7 & ? \\ 8 & ? \\ ? & ? \end{bmatrix}
• The second row is [9 \quad 10], which becomes the second column, so we can fill in the second column:
  • \begin{bmatrix} 7 & 9 \\ 8 & 10 \\ ? & ? \end{bmatrix}
• The third row is [11 \quad 12], which becomes the third column, so we can also fill in the third column:
  • \begin{bmatrix} 7 & 9 & 11 \\ 8 & 10 & 12 \end{bmatrix}
• Answer: \mathbf{B}^T = \begin{bmatrix} 7 & 9 & 11 \\ 8 & 10 & 12 \end{bmatrix}

Find the transpose of \mathbf{C} = \begin{bmatrix} 13 & 14 & 15 \\ 16 & 17 & 18 \end{bmatrix}

• The first row is [13 \quad 14 \quad 15], which becomes the first column, so the matrix will look like this so far:
  • \begin{bmatrix} 13 & ? \\ 14 & ? \\ 15 & ? \end{bmatrix}
• The second row is [16 \quad 17 \quad 18], which becomes the second column, so we can fill in the second column:
  • \begin{bmatrix} 13 & 16 \\ 14 & 17 \\ 15 & 18 \end{bmatrix}
• Answer: \mathbf{C}^T = \begin{bmatrix} 13 & 16 \\ 14 & 17 \\ 15 & 18 \end{bmatrix}

Find the transpose of \mathbf{D} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

• The first row is [a \quad b], which becomes the first column, so the matrix will look like this so far:
  • \begin{bmatrix} a & ? \\ b & ? \end{bmatrix}
• The second row is [c \quad d], which becomes the second column, so we can fill in the second column:
  • \begin{bmatrix} a & c \\ b & d \end{bmatrix}
• Answer: \mathbf{D}^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}

Find the transpose of \mathbf{E} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}

• The first row is [1 \quad 2], which becomes the first column, so the matrix will look like this so far:
  • \begin{bmatrix} 1 & ? \\ 2 & ? \\ ? & ? \end{bmatrix}
• The second row is [3 \quad 4], which becomes the second column, so we can fill in the second column:
  • \begin{bmatrix} 1 & 3 \\ 2 & 4 \\ ? & ? \end{bmatrix}
• The third row is [5 \quad 6], which becomes the third column, so we can fill in the third column:
  • \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}
• Answer: \mathbf{E}^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}

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