Question

What would the code be to test whether an array is two-dimensional? For one dimension I know that reversing the list will work. For two-dimensions I know that the opposite row / column must be the same. In other words [1][2] must equal [2][1] and so forth.

(defun symmetric-check (list) (equal list (reverse list)))

Answer 1

It depends on your definition of symmetry.

In linear algebra, a matrix is called symmetric iff it is equal to its transpose (this is equivalent to saying that M[i, j] = M[j, i] for all i and j). Thus,

(defun matrix-symmetric-p (m)
  (equal m (transpose-matrix m)))

(defun transpose-matrix (m)
  ;; implement this
  ...)

I strongly recommend using actual arrays, though, because it will make doing things like this easier and much more efficient.

(defun matrix-symmetric-p (m)
  (loop for i from 0 below (array-dimension m 0)
        always
          (loop for j from 0 below (array-dimension m 1)
                always
                  (= (aref m i j) (aref m j i)))))