Matrix Transpose

What Is a Matrix Transpose?

A matrix transpose is a new matrix obtained by interchanging the rows and columns of the original matrix. The elements of the rows become the elements of the columns, and conversely, the elements of the columns become the elements of the rows.

If we have a matrix $A$, then the transpose of matrix $A$ is usually denoted by $A^T$ or $A'$.

Formally, if matrix $A$ has an order of $m \times n$ with elements $a_{ij}$ (element in the $i$-th row and $j$-th column), then its transpose, $A^T$, will have an order of $n \times m$ with elements $a_{ji}^T = a_{ij}$.

This means that the element in the $j$-th row and $i$-th column of $A^T$ is the same as the element in the $i$-th row and $j$-th column of $A$.

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:

Notice how the first row $\begin{bmatrix} a & b \end{bmatrix}$ becomes the first column $\begin{bmatrix} a \\ b \end{bmatrix}$, and the second row $\begin{bmatrix} c & d \end{bmatrix}$ becomes the second column $\begin{bmatrix} c \\ d \end{bmatrix}$.

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

Some important properties of matrix transpose are:

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. $A = \begin{bmatrix} 1 & 3 & -5 \end{bmatrix}$
2. $B = \begin{bmatrix} 9 & -1 \\ 3 & 0 \\ 1 & 5 \end{bmatrix}$
3. $C = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 6 & -2 \\ 4 & 5 & -7 \end{bmatrix}$

Answer Key

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

   $A^T$ is a column matrix.

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

   $B^T$ is a rectangular matrix (horizontal matrix).

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

   $C^T$ is a square matrix.