Is a Matrix Symmetric if Row Space Equals Column Space?

[chaotixmonjuish]

I had this question on a test and I was wondering why it is false:

If the row space equals teh column space then A^T=A.

 

[slider142]

Consider the matrix
\left(\begin{array}{cc}1 & 1\\0 & 1\end{array}\right).

 

[chaotixmonjuish]

Ok, so I had a rationale why it was false but I'm not sure if I am close.

Obviously here the A^T does not equal A, but the dimension of the row space equals column space equals 2. Is this the right reasoning?

 

[slider142]

Ok, so I had a rationale why it was false but I'm not sure if I am close.

Obviously here the A^T does not equal A, but the dimension of the row space equals column space equals 2. Is this the right reasoning?

Well, the row space and the column space are both R², but the matrix is not symmetric.