In this article, we will define the transpose of a matrix along with properties of the transpose of a matrix using some examples.

##### Transpose of a matrix:

Transpose of a matrix A is obtained by changing its rows into columns and its columns into rows. It is written as A^{T}. Thus, in general, we can say that if we have a matrix ‘A’ of order (m×n) then its transpose A^{T} will be of order (n×m), where “m” is the number of rows and “n” is the number of columns. You can calculate the transpose of any matrix online.

Below we will take some examples of the transpose of different types of matrices along with examples. After that, we will learn about the various properties of the transpose of a matrix.

##### Examples of the transpose of a matrix

Example 1: Transpose of a 2×2 matrix is a 2×2. e.g.

Example 2. Transpose of a 3×3 matrix is a 3×3. An example of the transpose of a 3×3 matrix is;

Example 3. Transpose of a column matrix is a row matrix. Below is an example;

Example 4. Transpose of a diagonal matrix is the diagonal matrix itself. Below is an example;

Example 5. Transpose of a 2×3 matrix is a 3×2 matrix. Below is an example;

##### Properties of the transpose of a matrix

- (A
^{T})^{T}= A, where A is a matrix of order (m×n). - (A+B)
^{T}= (A^{T}) + (B^{T}), where A and B are the two matrices of the same order. - (KA
^{T}) = K(A^{T}), where K is a scalar which may be real or complex. - (AB)
^{T}= B^{T}A^{T}, where AB, is the product of two matrices A & B according to the properties of matrix multiplication.

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