Is a matrix that are equal to the identity matrix symmetric?

Question

If we have a matrix lets say $$ A= \pmatrix{ 2 & 2 & 6\\ 4 & 3 & 7\\ 9 & 2 & 2\\} $$ and reduce the matrix to the identity matrix $$ H= \pmatrix{ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\} $$ Can we then say that the matrix $A$ is symmetric because $H$ equals the identity matrix that are symmetric?

Thanks in advance!

Answer 1

A symmetric matrix is a square matrix A that satisfies $A=A^{T}$ when $A^{T}$ denotes the transpose of A, i.e $a_{ij}=a_{ji}$. Clearly the identity matrix is always symmetric. However as Luis said, symmetry doesn't preserve with elemental operations, and in fact $A=A^{T}$ is not true for your matrix A.

Answer 2

Every invertible matrix reduces to the identity matrix. There are plenty of invertible matrices that aren't symmetric.