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URL: https://nakafa.com/en/subjects/mathematics/matrix/matrix-transpose
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Learn matrix transpose by swapping rows and columns, then use its properties in worked examples.

---

## 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 <InlineMath math="A'" />.

Visible text: If we have a matrix , then the transpose of matrix is usually denoted by or <InlineMath math="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 $$a_{ji}^T = a_{ij}$$.

Visible text: Formally, if matrix has an order of with elements (element in the -th row and -th column), then its transpose, , will have an order of with elements .

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 $$A$$.

Visible text: This means that the element in the -th row and -th column of is the same as the element in the -th row and -th column of .

## 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.

Visible text: 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 $$A$$:

Visible text: Suppose we have matrix :

```math
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
```

Then, the transpose of matrix $$A$$ is:

Visible text: Then, the transpose of matrix is:

```math
A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}
```

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 $$\begin{bmatrix} c \\ d \end{bmatrix}$$.

Visible text: Notice how the first row becomes
the first column , and
the second row becomes
the second column .

### Matrix with Different Order

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

Visible text: Given matrix with order :

```math
B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
```

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

Visible text: The transpose of matrix , denoted , will have order :

```math
B^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}
```

- 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 $$B^T$$.

Visible text: - The first row of () becomes the first column of .
- The second row of () becomes the second column of .

### Column Matrix to Row Matrix

If $$C$$ is a column matrix:

Visible text: If is a column matrix:

```math
C = \begin{bmatrix} 2 \\ 3 \end{bmatrix}
```

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

Visible text: Then its transpose, , is a row matrix:

```math
C^T = \begin{bmatrix} 2 & 3 \end{bmatrix}
```

### Transpose of a Square Matrix

Given a square matrix $$D$$:

Visible text: Given a square matrix :

```math
D = \begin{bmatrix} -7 & 8 & 1 & 3 \\ 5 & 9 & 7 & 2 \\ 2 & 2 & 1 & 3 \\ 1 & -6 & 0 & 1 \end{bmatrix}
```

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

Visible text: Then its transpose, , is also a square matrix:

```math
D^T = \begin{bmatrix} -7 & 5 & 2 & 1 \\ 8 & 9 & 2 & -6 \\ 1 & 7 & 1 & 0 \\ 3 & 2 & 3 & 1 \end{bmatrix}
```

## 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)

Visible text: 1. (The transpose of a transpose matrix is the
 matrix itself)
2. (Transpose of the sum of two matrices)
3. (Transpose of the subtraction of
 two matrices)
4. , where is a
 scalar
5. (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.  $$C = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 6 & -2 \\ 4 & 5 & -7 \end{bmatrix}$$

Visible text: 1. 
2. 
3.

### 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.

Visible text: 1. 

 is a column matrix.

2. 

 is a rectangular matrix (horizontal matrix).

3. 

 is a square matrix.