# Transpose of a Matrix

*Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices. In this article, we will briefly discuss how to transform a matrix, its properties and examples.*

This article will cover everything you need to know about transposing a matrix and how to do it.

Let’s get started!

**Table of Content**

- What does Transposing a Matrix mean?
- How to Transpose a Matrix in Python
- Properties and Theorem of Transpose of a Matrix

**What Does Transposing a Matrix Mean?**

When we talk about transposing a matrix, it means switching the rows and columns of that matrix. Technically, you are not changing the matrix itself but the location of its elements. Transposing a matrix changes its size but not its values.

**Definition**

Transpose of a matrix is a matrix that is obtained by interchanging the rows and columns.

or

Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices.

**Representation**

If A is any matrix, then Transpose of A is given by **A ^{T}** or

**A’.**

**i.e., if A is any matrix of order m by n, then the order of transpose of A will be n by m.**

**A** = [a_{ij}]_{m x n}, then **A ^{T}** =

**A’**= [a

_{ij}]

_{n x m}.

**Example**

Here, we have taken two matrices, A and B, of order 3 x 4 and 3 x 3, respectively, and when we transpose them, the order of matrices will be 4 x 3 and 3 x 3, respectively.

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**Also Read:** All about Symmetric Matrix

**How to Transpose a Matrix using Python**

In the above examples of transposing a matrix, we have seen that transposing a matrix is nothing but inter-changing the rows and columns (or changing the row to the column and vice versa) of a matrix.

Now, we will see how to transform matrices using Python.

**Using NumPy**

**Code**

#Python Programm to Transpose a Matrix using Numpy# import numpy import numpy as np #create a matrix using arraymatrix = np.array ([[1, 2, 3], [2, 4, 6], [7, 8, 9]]) #using transform function of numpymatrix.transpose()

**Output**

2. **Using Nested Loop**

#Python Programm to Transpose a Matrix using Nested Loop #create a matrixA = [[0, 2, 3], [-2, 0, 6], [-3, -6, 0], [8, 6, 5]] #create a matrix to store the result T = [[0,0,0,0], [0,0,0,0], [0,0,0,0]] # iterate through rowsfor x in range(len(A)): #iterate through columns for y in range(len(A[0])): T[y][x] = A[x][y] for t in T: print(t)

**Output**

**Properties of Transpose of a Matrix**

**Transpose of Transpose of a Matrix**

Transpose of Transpose of a matrix is again the same matrix, i.e., for any matrix A.

**(A ^{T})^{T} = A**

**Example**

**Addition**

If A and B are two matrices, then the transpose of A + B is equal to the transpose of A + transpose of B, i.e.,

**(A + B) ^{T} = A ^{T} + B ^{T} **

**Example**

**Scalar Multiplication**

If a matrix is multiplied by a constant and then transpose is taken, then the result is equal to the transpose of the original matrix multiplied by a constant, i.e.,

**(kA) ^{T} = kA ^{T} **

**Example**

**Multiplication**

If A and B are two matrices, then the transpose of AB is equal to the product of the transpose of A and transpose of B, i.e.,

**(AB) ^{T} = B ^{T} A^{T}**

**Example:**

**Inverse**

If A is an invertible matrix, then the transpose of the inverse of A is equal to the inverse of the transpose of A, i.e.,

**(A^{-1})^{T} = (A^{T})^{-1}**

**Example:**

**Determinant**

For any square matrix A, the determinant of A is equal to the determinant of the transpose of A, i.e.,

**det (A) = det(A ^{T})**

**Example**

Also Read: All About Skew-Symmetric Matrix

**Conclusion**

In this article, we have briefly discussed the transpose of a matrix with examples in Python. The process of transposing a matrix is simple; the most important thing to remember is that you must interchange rows and columns.

Hope this article will help you to learn the concepts of transposing a matrix.

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## FAQs on Transpose of Matrix

**What is the Transpose of a Matrix?**

Transpose of a matrix is a matrix that is obtained by interchanging the rows and columns.

or

Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices.

**How do you find the transpose of a matrix?**

To find the transpose of a matrix, rewrite the first row of the matrix as the first column, the second row as the second column, and so on for all rows.

**What is the transpose of a rectangular matrix?**

The transpose of a rectangular matrix is another rectangular matrix where the number of rows and columns are interchanged. So, if the original matrix has dimensions

, the transpose will have dimensions .**How does transposing affect the determinant of a matrix?**

The determinant of a matrix remains unchanged upon transposing. That is,

.**Can the transpose of a matrix change its rank?**

No, the rank of a matrix is invariant under transposition. This means that the rank of

is equal to the rank of .**About the Author**

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio