Transpose Matrix Calculator

Now transform matrices with the transpose matrix calculator and simplify complex calculations of transpose matrix dimensions to streamline your mathematical tasks.

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Introduction to Transpose Matrix Calculator:

Transpose matrix calculator is an online tool that helps you to evaluate the transpose of the given matrix. Our tool is used to find the transpose of a matrix that has different orders like 2 by 2 and 3 by 3.

Transpose Matrix Calculator with Steps

Although transpose is an easy method when you solve a 2 by 2 matrix, when you go for a larger order of matrix transposition, the calculation becomes complex when done manually. In this situation, you need the matrix transpose calculator that gives the transpose of larger matrices.

What is a Transpose of a Matrix?

Transpose of a matrix is a process in linear algebra that is used to form a new matrix after converting the row into a column and the column into a row. It is denoted as A^T for n × m order of matrix. The dimension of a matrix is changed from n × m to m × n after transposition.

You can use this method to solve the basic operation like addition or multiplication after finding the transpose of a matrix. The formula used by the transpose matrix calculator is,

$$ A^T \;=\; [a_{ij}]^T \;=\; [a_{ji}] $$

How to Find Transpose of a Matrix?

To find the transpose of a matrix, you need to get familiar with the basis of the matrix. So you can find out the transpose of the matrix for addition and multiplication operations. Let us know how the transpose of a matrix calculator finds the transpose of a matrix in steps.

Step 1:

Identify the matrix A with dimensions m × n.

Step 2:

For the transpose of a matrix A, convert the row into and column into a row.

Step 3:

After converting a row into a column and a column into a row the dimension of matrix A m × n to the dimensions n × m.

Step 4:

Now you get the transposition of the matrix, express the matrix A as A^T that shows the transposition

Let's see the example of a transpose matrix with a solution where all the calculation steps are included.

Suppose a 2 by 2 matrix to find the transpose,

$$ A \;=\; \left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right] $$

Solution:

Here, a square matrix A is a 2×2 matrix, so A^T will be a 2×2 matrix.

For the transpose of a matrix, the first column of A^T is the first row of A; the second column of A^T is the second row of A. Therefore,

$$ A \;=\; \left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right] $$

How to Use Transpose Matrix Calculator?

Matrix transpose calculator has an easy-to-use interface so you just need to enter your problem of matrices transpose to get a solution in a simplified method. Follow our guidelines before using it . These guidelines are:

Select the matrix size (2 by 2 or maybe 3 by 3) from the given list for the transpose matrix.
Enter the elements of the given matrix transpose in the next input box.
Review the given matrix value before hitting the calculate button to start the evaluation process in the transpose of a matrix calculator.
Click the “Calculate” button to get the solution of your given matrix transpose problem
If you want to try out our transpose of matrix calculator for the first time then you must try out the load example to learn more about it.
Click on the “Recalculate” button to get a new page for finding more example solutions of transpose matrix problems.

Output of Matrix Transpose Calculator

Transpose matrix Calculator gives you the solution to a given matrix question when you add the input into it. It may be included as:

Result option

When you click on the result option it gives you a solution of transpose matrix.

Possible steps

It provides you with a solution of matrix transpose where all the calculations of the square matrix in the steps that are mentioned.

Useful Features of Transpose of a Matrix Calculator

The matrix transpose calculator provides several features that you use to calculate the transposition of matrices problems and provides solutions immediately. These features are:

It is a free tool that enables you to evaluate it for free to find the matrix transpose problem in the solution.
It is a manageable tool that can solve different orders of matrices to find the transpose.
Our transpose of matrix calculator helps you to get a strong hold on the transpose method concept for matrix when you use it for practice more examples.
It saves the time that you consume on the calculation of the transpose matrix in a couple of minutes.
It is a reliable tool that provides precise solutions whenever you use it to calculate the transposition without any error.
Transpose matrix Calculator provides the solution without sign-in condition so you can use it anywhere.

Related References

How would you define transpose matrix?
Explain the evaluation process of matrix transpose:
Give some examples of matrix transpose:
Give an overview of matrix transpose?

Frequently Ask Questions

Does a matrix and its transpose have the same determinant?

No, a matrix and its transpose do not have the same determinant. In a special case for square matrices, the determinant remains the same after taking the transpose.

For a square matrix A, where the number of rows is equal to the number of columns. If A is symmetric A = A^T.

Then the determinant of A is equal to the determinant of its transpose. It holds the property because the transpose of a symmetric matrix is itself. Such as:

$$ det(A) \;=\; det(A^T) $$

What is the inverse of a transpose matrix?

The inverse of a transpose matrix is equal to the transpose of the inverse of the original matrix, provided that the original matrix is invertible because it has a non-zero determinant. If A is an invertible matrix,

$$ (AT)^{−1} \;=\; (A^−1)T $$

This property is useful in various matrix operations and calculations when dealing with systems of linear equations and transformations.

What happens when a matrix is multiplied by its transpose?

When a matrix is multiplied by its transpose, the resulting matrix is a symmetric matrix. If A is an m × n matrix, then A^T is an n × m matrix. When you multiply A by its transpose (A^T), you get an m × m symmetric matrix.

$$ Symmetric\; Matrix\; A \;=\; A \times A^T $$

This symmetric matrix has the property that its entries below and above the main diagonal are equal. This leads to bij = bji for all i,j which is a symmetric matrix.

This operation is also sometimes used in inner products in vector spaces, where it plays a significant role in defining norms and angles between vectors.

Does a matrix and its transpose have the same eigenvalues?

Yes, the matrix and its transpose have the same eigenvalues. If A is a square matrix and λ is an eigenvalue of A with corresponding eigenvector x. Then A^T also has the same eigenvalue λ with the same eigenvector x.

$$ A ⋅ x \;=\; λ ⋅ x $$

This property holds for any square matrix, whether it is symmetric or not. However, for symmetric matrices, they have the same eigenvalues as their transposes.

What is the transpose of a diagonal matrix?

When you take the transpose of a diagonal matrix, the result is a matrix itself. A diagonal matrix is a square matrix where all the elements other than the diagonal are zero.

For the transpose of a matrix, the rows become columns and columns become rows. Since there are no of diagonal elements to change positions, the resulting matrix remains the same. Mathematically, if D is a diagonal matrix, then

$$ D^T \;=\; D $$

Amanda Jepson

Last Updated February 21, 2024