<prompt> HebumathbfM anzapmatrix a b c b c a c a b endpmatrixkuwa tumbo lenye maingizo changamano kiasi kwamba mathbf M2 mathbfI. Ikiwa abc 1, basi tafuta thamani zinazowezekana za a3 b3 c3.  <response>: Tunapata hilomathbfM2 anzapmatrix a b c b c a c a b mwishopmatrix anzapmatrix a b c b c a c a b mwishopmatrix anzapmatrix a2 b2 c2 ab ac bc ab ac bc ab ac bc a2 b2 c2 ab ac bc ab ac bc ab ac bc a2 b2 c2 mwisho pmatrix.Kwa kuwa hii ni sawa na mathbfI, tunaweza kusema kwamba a2 b2 c2 1 na ab ac bc 0.Kumbuka factorizationa3 b3 c3 - 3abc a b ca2 b2 c2 - ab - ac - bc.Tuna hiyoa b c2 a2 b2 c2 2ab ac bc 1,hivyo a b c pm 1.Ikiwa a b c 1, basia3 b3 c3 - 3abc a b ca2 b2 c2 - ab - ac - bc 1,hivyo a 3 b3 c3 3abc 1 4.Ikiwa a b c -1, basia3 b3 c3 - 3abc a b ca2 b2 c2 - ab - ac - bc -1,hivyo a3 b3 c3 3abc - 1 2.Kwa hivyo, thamani zinazowezekana za a3 b3 c3 ni 2,4. 