Transpose of a Matrix

# Transpose of a Matrix

Vikram Singh
Assistant Manager - Content
Updated on Apr 12, 2024 10:56 IST

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.

In math, transposing a matrix is the process of switching its columns and rows. Transposing a matrix changes the location of its elements but doesn’t alter their properties. A transposed matrix is one that has been switched from its original position.

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

Let’s get started!

## 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 AT 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 = [aij]m x n, then AT = A’ = [aij]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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## 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.

1. 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()Copy code`

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)Copy code`

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.

(AT)T = A

Example

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 AT

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 = (AT)-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(AT)

Example

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## 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.

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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 .