The vector x must be (1 x n); the matrix A must be (n x m); the vector b must be (1 x m).
If you take the transpose of both sides, you get:
(xA)^T = b^T
Rearranging the LHS:
(A^T)(x^T) = b^T
Now A^T is an (m x n) matrix; x is a (n x 1) vector; b is an (m x 1) vector.
If A is square and symmetric, then by definition A^T = A. No work needed.
You can solve for x^T = (A^T)^-1 (b^T)
using the usual techniques.
I would not recommend computing a matrix inverse. If your matrix is square, you're better off using LU decomposition and forward-back substitution instead. It's far more stable.