Matrix Transpose

Turning Matrix Rows into Columns

A matrix transpose is formed by swapping rows and columns. If entry $a_{ij}$ is in row $i$ and column $j$ of $A$, the same entry is in row $j$ and column $i$ of $A^T$.

The transpose of $A$ is written as $A^T$ or $A'$. If $A$ has dimensions $m \times n$, then $A^T$ has dimensions $n \times m$ and satisfies $(A^T)_{ji} = a_{ij}$.

How to Determine the Matrix Transpose

To obtain the matrix transpose, follow these steps:

1. Write the first row of the original matrix as the first column of the transpose matrix.
2. Write the second row of the original matrix as the second column of the transpose matrix.
3. Continue this process for all rows in the original matrix.

General Matrix

Suppose we have matrix $A$:

Then, the transpose of matrix $A$ is:

The first row $\begin{bmatrix} a & b \end{bmatrix}$ becomes the first column:

The second row $\begin{bmatrix} c & d \end{bmatrix}$ becomes the second column:

Matrix with Different Order

Given matrix $B$ with order $2 \times 3$:

The transpose of matrix $B$, denoted $B^T$, will have order $3 \times 2$:

• The first row of $B$ ($\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$) becomes the first column of $B^T$.
• The second row of $B$ ($\begin{bmatrix} 4 & 5 & 6 \end{bmatrix}$) becomes the second column of $B^T$.

Column Matrix to Row Matrix

If $C$ is a column matrix:

Then its transpose, $C^T$, is a row matrix:

Transpose of a Square Matrix

Given a square matrix $D$:

Then its transpose, $D^T$, is also a square matrix:

Properties of Matrix Transpose

The transpose satisfies the following identities:

1. $(A^T)^T = A$ (The transpose of a transpose matrix is the matrix itself)
2. $(A + B)^T = A^T + B^T$ (Transpose of the sum of two matrices)
3. $(A - B)^T = A^T - B^T$ (Transpose of the subtraction of two matrices)
4. $(kA)^T = kA^T$, where $k$ is a scalar
5. $(AB)^T = B^T A^T$ (Transpose of the product of two matrices, note the reversed order)

Exercises

Determine the transpose of the following matrices and state the type of the resulting matrix (e.g., row matrix, column matrix, square matrix).

1. Matrix $A$:

   $$A = \begin{bmatrix} 1 & 3 & -5 \end{bmatrix}$$

2. Matrix $B$:

   $$B = \begin{bmatrix} 9 & -1 \\ 3 & 0 \\ 1 & 5 \end{bmatrix}$$

3. Matrix $C$:

   $$C = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 6 & -2 \\ 4 & 5 & -7 \end{bmatrix}$$

Worked Solutions

1. The transpose of $A$ is:

   $$A^T = \begin{bmatrix} 1 \\ 3 \\ -5 \end{bmatrix}$$

   The single row of $A$ becomes a single column. Thus, $A^T$ has order $3 \times 1$ and is a column matrix.

2. The transpose of $B$ is:

   $$B^T = \begin{bmatrix} 9 & 3 & 1 \\ -1 & 0 & 5 \end{bmatrix}$$

   Each row of $B$ becomes a column of $B^T$. The dimensions change from $3 \times 2$ to $2 \times 3$, so the result is a rectangular matrix.

3. The transpose of $C$ is:

   $$C^T = \begin{bmatrix} 2 & 1 & 4 \\ 3 & 6 & 5 \\ 1 & -2 & -7 \end{bmatrix}$$

   Reading the columns of $C$ as the rows of $C^T$ gives the matrix above. Its order remains $3 \times 3$, so it is still a square matrix.