Question

Is the transpose of a matrix symmetric?

Answer 1

Checking if a transpose is symmetric:

Transpose of a matrix is the matrix obtained by interchanging the rows and columns. So, the first row becomes the first column, the second row becomes the second column, and so on.

The transpose of a matrix is denoted by $A^{T.}$

A symmetric matrix is a square matrix in which the transpose matrix is the same as the original matrix that is: $A^T=A$.

Consider matrix:

$A=\left[\begin{array}{ccc}1& 1& -1\\ 1& 2& 7\\ -1& 0& 5\end{array}\right]$

On taking the transpose of the matrix, we have

$A^T=\left[\begin{array}{ccc}1& 1& -1\\ 1& 2& 0\\ -1& 7& 5\end{array}\right]$

Since, $A^T\ne A$, the matrix is not symmetric.

Therefore, we can say that the transpose of a matrix is not necessarily symmetric.