Question

Is a transpose a symmetric?

Answer 1

From the question we are to check whether a transpose matrix is symmetric or not. For solving this question we will use the concept of matrices. We will use the definition of symmetric matrices and explain the transpose of a matrix. Thus, we solve the given question.

Complete step-by-step solution:
Generally, in matrices if we want to calculate the transpose of a matrix we will simply interchange the row elements and column elements of the given matrix among each other i.e. we write the elements of the rows as columns and write the elements of the rows.
In matrix algebra a given matrix is said to be a symmetric matrix when it obeys the statement mentioned belows.
A square matrix is said to be a symmetric matrix if the transpose of the matrix is the same as the given matrix.
So, from the above definitions of the transpose and symmetric matrices we can conclude as follows.
No, the transpose of a matrix is not symmetric. If the rows and columns of a matrix are interchanged such that,
The first row becomes the first column and the second row becomes the second column, and so on, then it is the transpose matrix.
A symmetric matrix is a square matrix which is symmetrical about its diagonal.
From this we can say that, if A is any symmetric matrix, then $A={{A}^{T}}$.

Note: Students should have good knowledge in the concept of matrix algebra. We must know the definitions and properties of symmetric matrix and transpose of a matrix.
The important point to be noted is that if A is any symmetric matrix, then $A={{A}^{T}}$ but the transpose of the matrix is not symmetric.