You should use the columns of the matrix for the arrows, not the rows:
A = {{1, 1, 1}, {1, 6, 8}, {1, 8, 9}};
c = Orthogonalize[Eigenvectors[A]] // N;
c // MatrixForm
Graphics3D[{GeometricTransformation[Cuboid[], c],
Map[Arrow[{{0, 0, 0}, #}] &, Transpose[1.5 c]]}, Axes -> True]
or else
Graphics3D[{GeometricTransformation[{Cuboid[],
Map[Arrow[{{0, 0, 0}, #}] &, 1.5 IdentityMatrix[3]]}, c]},
Axes -> True]