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 2 by 2 and 3 by 3 orders.

Transpose Matrix Calculator with Steps

Although transpose is an easy method, so you can solve matrix (2 x 2), but when it comes to larger order of matrix transposition, the calculation becomes complex. In this situation, you need the matrix transpose calculator that can determine the transpose of larger matrices.

What is a Transpose of a Matrix?

Transpose of a matrix is a process in linear algebra, used to form a new matrix by 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 multiplication or addition 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 basics of matrix. You can find the addition matrix transpose and multiplication operations. Let us know how the transpose of a matrix calculator finds the 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 a column 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 will be converted to dimensions n × m.

Step 4:

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

Let's see the example of transpose matrix with a solution to understand the calculations in steps.

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 matrix, the first column of A^T becomes the first row of A; the second column of A^T becomes 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 matrices transpose question to get a solution. Follow our guidelines before using it.

Select the matrix size (2 by 2 or maybe 3 by 3) from the given list to get 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 of transpose of a matrix calculator.
Click the “Calculate” button to get the solution of your given matrix transpose problem.
If you are trying our transpose of matrix calculator for the first time then you must use the load example option.
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 solution of 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 the step by step solution of matrix transpose in detail.

Advantages of Transpose of a Matrix Calculator:

The matrix transpose calculator provides benefits that you use to calculate the transposition of matrices problems. These advantages are:

It is a free tool that enables you to find the matrix transpose problem.
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.
It saves the time that you consume in calculation of transpose matrix.
It is a reliable tool that calculate the transposition matrix and gives accurate solutions.
Transpose matrix calculator gives the solution without sign-in condition so you can use it anywhere and anytime.

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. Here's the formula to find the eigen value:

$$ A ⋅ x \;=\; λ ⋅ x $$

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 diagonal elements to change positions, the resulting matrix remains the same. Mathematically, if D is a diagonal matrix, then you can find diagonal matrix as:

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

Amanda Jepson

Last Updated February 21, 2024