Matrix Transpose Calculator: Find Transpose of a Matrix

This matrix transpose calculator will help you to find the transpose of the matrix containing either real numbers, complex numbers, or both. So let’s go through the article below that is specially arranged to get a sound knowledge of matrix transpose. Go ahead!

What Is Matrix Transpose?

In the context of mathematics:

“The matrix obtained by interchanging the rows and columns with each other is known as the matrix transpose”

Generic Expression:

If $$ A = \begin{bmatrix} a&b \\ c&d \\\end{bmatrix} $$ Then $$ A = \begin{bmatrix} a&c \\ b&d \\\end{bmatrix} $$

Methodology:

You can either use the best transpose of a matrix calculator to have an instant idea of the methodology. However, we will pack that guide right here for you people. So let’s get through them!

If a matrix A has order as $A_{i*j}$, then its transpose is obtained if you change the order as $A_{j*i}$
By taking transpose of a matrix, its dimensions also change

Properties of Matrix Transpose:

Make a supposition that you are having a couple of matrices named as A and B along with a constant x. Are you done with that? Great! Let’s move now toward discussing the properties of the matrix transposition!

Property # 01:

$$ \left(A^{t}\right)^{t} = A $$

Property # 02:

$$ \left(A +B\right)^{t} = A^{t} + B^{t} $$

Property # 03:

$$ \left(AB\right)^{t} = B^{t} A^{t} $$

Property # 04:

$$ \left(xA\right)^{t} = xA^{t} $$

Property # 05:

$$ det\left(A^{t}\right) = det A $$

Property # 06:

$$ \left(A^{-1}\right)^{t} = \left(A^{t}\right)^{-1} $$

Property # 07:

If there is a square matrix, then its eigen values would be exactly equivalent to the eigenvalues of transpose. Our free conjugate transpose calculator also satisfies all of these characteristics to display the actual transpose of the matrix.

How To Find The Transpose of a Matrix?

This is where we are going to resolve a problem to throw a more brighter light on the concept so that you could get it better. Stay focused!

Example # 01:

Find the transpose of the matrix given as under: $$ A = \begin{bmatrix} 3&5 \\ 7&9 \\\end{bmatrix} $$

Solution:

Flipping the rows and columns with each other: $$ A^{t} = \begin{bmatrix}3&5\\7&9\\\end{bmatrix} ^ \text{t} $$ $$ \begin{bmatrix}3&7\\ 5&9 \\\end{bmatrix} $$ Which is our desired matrix. If you think of any errors in calculations, you can cross check them by using this matrix transpose calculator.

Example # 02:

How to transpose a matrix which is in 6 by 7 order and is as follows: $$ B = \begin{bmatrix} 4&3&8&-9&10&2&8\\10&9&7&3&8&-8&-8\\1&7&8&9&1&1&7 \\ 8&9&4&8&-8&6&1\\1&11&6&1&2&6&-4\\-1&7&2&8&6&4&4\\\end{bmatrix} $$

Solution:

Moving forward to find transpose of a matrix: $$ B^{t} = \begin{bmatrix} 4&3&8&-9&10&2&8\\10&9&7&3&8&-8&-8\\1&7&8&9&1&1&7 \\ 8&9&4&8&-8&6&1\\1&11&6&1&2&6&-4\\-1&7&2&8&6&4&4\\\end{bmatrix}^{t} $$ B^{t} = \begin{bmatrix}4&10&1&8&1&-1\\3&9&7&9&11&7\\8&7&8&4&6&2\\-9&3&9&8&1&8 \\10&8&1&-8&2&6\\ 2&-8&1&6&6&4\\ 8&-8&7&1&-4&4\\\end{bmatrix} $$

How Matrix Transpose Calculator Works?

Now you could instantly find transpose of a matrix by tapping a couple of clicks while using this free matrix transformation calculator. Let’s see how it could be done!

Input:

First, you need to set the order of the matrix. This could be done by selecting the number of rows and columns from a couple of drop-down lists available
After you do that, it’s time to get the desired layout of the matrix by tapping on “Set Matrices”
Now, write down entities in their designated fields
At last, hit the calculate button

Output: The best transpose matrix calculator determines:

The transformed version of the given matrix

FAQ’s:

What is the product of a matrix and its transpose?

If you multiply the matrix a matrix with its transpose, you will always get an identity matrix. For further assistance, you can subject to our free online matrix multiplication calculator for verification.

Does a matrix commute with its transpose?

Usually an upper triangular matrix does commute with its transpose. You can also verify it by using the online matrix transpose calculator.

Why is a transpose a symmetric?

Transpose is a symmetric property as a calculating product of matrices and its transpose yields an identity matrix.

Why do we use matrix transpose?

We take transpose to introduce easiness in our computations. Just take an example of matrix multiplication. When two matrices are not ok forproduct, then taking their transpose will make it possible to calculate their multiplication.

Do all matrices have transpose?

Yes, you can interchange the rows and columns of any matrix, so all matrices have their transpose.

Conclusion:

Matrices provide assistance in resolving electrical circuit queries by applying various operations. Not only this, but you can analyse different particular quantum mechanics phenomena with the help of matrix techniques. Our free matrix transpose calculator makes it quite easy to flip the given matrix so as to make the calculations more easy and generate accurate results.

References:

From the source of Wikipedia: Transpose, Properties, Products, Transposes of linear maps and bilinear form, Adjoint From the source of Khan Academy.fandom: Matrix transpose, Strategies, Real-life Applications From the source of Lumen Learning: Row Operations and Augmented Matrices, System of Equations, Row Operations