gh-39876: Faster implementation of is_invertible() by checking full rank
    
#31274
As mentioned in the issue, a faster invertibility check for matrix in
`Matrix_gf2e_dense` by checking full rank.

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### ⌛ Dependencies

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URL: #39876
Reported by: Henry Wu
Reviewer(s): user202729, Vincent Neiger

Author: Release Manager

src/sage/matrix/matrix_gf2e_dense.pyx

@@ -1004,6 +1004,22 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
                 break
         return pivots
 
+    def is_invertible(self):
+        """
+        Return ``True`` if this matrix is invertible.
+
+        EXAMPLES::
+
+            sage: m = Matrix(GL(2^8, GF(2^8)).random_element())
+            sage: m.is_invertible()
+            True
+        """
+        if self._ncols != self._nrows:
+            return False
+        if self._nrows != self.rank():
+            return False
+        return True
+
     def __invert__(self):
         """
         EXAMPLES::
@@ -1021,7 +1037,7 @@ cdef class Matrix_gf2e_dense(matrix_dense.Matrix_dense):
         cdef Matrix_gf2e_dense A
         A = Matrix_gf2e_dense.__new__(Matrix_gf2e_dense, self._parent, 0, 0, 0)
 
-        if self.rank() != self._nrows:
+        if not self.is_invertible():
             raise ZeroDivisionError("Matrix does not have full rank.")
 
         if self._nrows: