Transpose of Matrix

Transpose of Matrix Meaning

The transpose of matrix A can be recognized as the matrix appeared by rearranging the rows as columns and columns as rows. As a result, the indices of each element are interchanged. Generally, the transpose of matrix A is defined as 

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Where,

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The diagonals of the matrix and transpose matrix remain unchanged but all the other elements are rotated around the diagonal. The size of the matrix changes from m×n to n×m.

The important properties of the transpose of matrices permit the manipulation of matrices in a simple manner. Also, some essential transpose matrices are defined on the basis of their features. The matrix will be considered as symmetric if the matrix is equivalent to its transpose. The matrix will be considered as skew- symmetric if the matrix will be equal to the negative of the transpose. The conjugate transpose of a matrix is the transpose of the matrix with the elements replaced with its complex conjugate.


What is the Matrix?

In Mathematics, the matrix is the rectangular ordering of numbers, symbols or expressions, arranged in rows and columns. Matrices are commonly expressed in brackets. The horizontal and vertical lines of elements in matrices are known as rows and columns. The size of a matrix is defined by the number of rows and columns present in them. A matrix with row p and column q is known as p×q matrix or p-by-q matrix, while p and q are dimensions of matrix. The dimensions of the matrix given below are 2×3( read as “ two by three”), as there are two rows and three columns.

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The individual items such as numbers, symbols, or expressions are known as its elements or entries. Provided that two matrices are of the same size (having the equal numbers of rows and columns), two matrices can be subtracted or added element by element. Two matrices can only be multiplied if the number of columns in the first matrices is equivalent to the number of rows in the second matrix.

Matrices with a single row are known as row vectors whereas matrices with a single column are known as column vectors. Matrix with the same number of rows and columns is known as a square matrix. A matrix with no rows and columns is known as an empty matrix.

How to find the transpose of a matrix?

Let us take two matrices P and Q. 

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Though both the matrices have a similar set of elements, are both the matrices equal?

No, the matrices shown above are not equal as their order is not the same. Now, it is important to observe that there can be multiple matrices which have exactly the same elements as P  

Here, the number of rows and columns in P  is equal to the number of columns and rows in Q respectively. Hence,  the matrix P is known as the transpose of the matrix Q. The transpose of matrix P is expressed  by P’ or PT

If P = [aij] m×n, then P’ = [aij] m×n, then

Hence, the transpose of a Matrix can be expressed as “A Matrix which is designed by organizing all the rows of a given matrix into columns and columns of given matrix into rows.

Let us understand how to find the transpose of a matrix through an example.


Transpose of Matrix Example

  1. Find the transpose of the following matrix

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Solution:  Given, the matrix of order 2* 3

The transpose of a matrix is obtained by interchanging the rows and columns of the given matrix.

Hence, the transpose of matrix for the above matrix is :

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Properties of Transpose of Matrices

Now, we will understand the transpose matrix by considering two matrices P and Q which are equal in order. Some of the properties of the transpose of matrices are mentioned below:

  1. Transpose of the Transpose Matrix

The transpose of the transpose of the matrix is the matrix itself = (PT)T =P

For example:

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  1. Addition Property of Matrices

The transpose of the addition of two matrices is similar to the addition of their transposes = (P + Q)T = P T + Q T

For Example = 

Proof - (P + Q)T = P T + Q T

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LHS = RHS

Hence, proved.


  1. Multiplication By Constant

The transpose of the multiplication of two matrices i.e P×Q is similar to the multiplications of transposes in the reverse order = (aP) T = aPT

For Example 

To prove: (aP) T = aPT

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LHS = RHS

Hence proved


  1. Multiplication Property Of Transpose

When we multiply a scalar matrix by the matrix, the order of the transpose we get is irrelevant (PQ) T = PT QT

For example:

To prove: (PQ) T = PT QT

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LHS = RHS

Hence Proved


Solved Examples

  1. Find the transpose of the following matrix

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Solution: We will name this matrix A. It has 2 rows and 3 columns which implies that AT will have 3 rows and 2 columns. Therefore, AT will be represented as

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 Where * indicates elements that must be calculated.

As the diagonal elements are not changed when transposing a matrix ,we emphasize this in the original matrix.

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And transform them into the transpose matrix as shown below.

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We will focus the first row in the original matrix

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And represent this as the first column of the transpose matrix:

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Further, we will focus the second row of the original matrix

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And write the elements in the order as the second column of matrix

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  1. Find the transpose of the given matrix:

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Solution: Given, the matrix of order 3* 3

The transpose of a matrix is obtained by interchanging the rows and columns of the given matrix.

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Facts

  • The Term Matrix Was Introduced By James Joseph Sylvester In 1850.

  • An English Mathematician Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously explained the initial important use of notation A = ai×jto denote a matrix where ai×j indicates the elements found in ith row and jth column.

FAQ (Frequently Asked Questions)

1. What is The Difference Between Transpose And Inverse Of A Matrix?

The difference between transpose and inverse of the matrix is given below:

  • Transpose of the matrix is received by rearranging the rows and columns in the matrix whereas the inverse is received by a relatively complex numerical calculation(but in reality both transpose and the inverse of the matrix are linear transformations).

  • As a direct outcome, the elements on the transpose only change their position but their values remain the same. But in the inverse of the matrix, the numbers may be entirely different from the original matrix.

  • Each and every matrix can have transpose, but the inverse is only stated for square matrices, and the determination has to be a non-zero determinant.

2. Why Transpose Matrices?

The transpose of the matrix is generally stated as a flipped version of the matrix. Transpose of the matrix can be done by rearranging its rows and columns. Transpose of matrix M is represented by MT

There are numerous ways to transpose matrices.The transpose of matrices is basically done because they are used to represent linear transformation. Calculating the transpose of a matrix that indicates some linear transformation can reveal some properties of transformation.

For example, If M is a matrix and its transpose is MT , then if MT = M we can state that the following matrix is symmetric and therefore it corresponds to symmetric transformation. It implies that if A is a transformation that operates two vectors i.e. x and y, then these two vectors can be replaced with each other and the result will be the same.