<prompt> Tafuta matrix mathbfM, yenye maingizo halisi, kama hayomathbfM3 - 4 mathbfM2 5 mathbfM anzapmatrix 10 20 5 10 mwishopmatrix.  <response>: Acha mathbfM anzepmatrix a b c d endpmatrix. Kumbuka kwambamathbfM mathbfM3 - 4 mathbfM2 5 mathbfM mathbfM4 - 4 mathbfM3 5 mathbfM2 mathbfM3 - 4 mathbfM2 5 mathbfM mathbfM,hivyoanzapmatrix a b c d mwishopmatrix anzapmatrix 10 20 5 10 mwishopmatrix anzapmatrix 10 20 5 10 mwishopmatrix anzapmatrix a b c d mwishopmatrix.Hii inakuwaanzapmatrix 10a 5b 20a 10b 10c 5d 20c 10d mwishopmatrix anzapmatrix 10a 20c 10b 20d 5a 10c 5d endpmatrix.Tukilinganisha maingizo, tunapataanzaalign10a 5b 10a 20c, 20a 10b 10b 20d, 10c 5d 5a 10c, 20c 10d 5b 10d.endalignKisha kutoka milinganyo ya kwanza na ya pili, 5b 20c na 20a 20d, hivyo b 4c na a d. Milingano nyingine inatupa taarifa sawa. Hivyo,mathbfM anzapmatrix a 4c c a mwishopmatrix.KishamathbfM2 anzapmatrix a 4c c a mwishopmatrix anzapmatrix a 4c c a endpmatrix anza pmatrix a2 4c2 8ac 2ac a2 4c2 mwishopmatrix,namathbfM3 anzapmatrix a 4c c a mwishopmatrix anzapmatrix a2 4c2 8ac 2ac a2 4c2 mwishopmatrix anzapmatrix a3 12ac2 12a2 c 16c3 3a2 c 4c3 a3 12ac2 mwisho pmmatrix.Kwa hivyo,anzaalignmathbfM3 - 4 mathbfM2 5 mathbfM anzapmatrix a3 12ac2 12a2 c 16c3 3a 2 c 4c3 a3 12ac2 mwishopmatrix - 4 anzapmatrix a2 4c2 8ac 2ac a2 4c2 mwishopmatrix 5 anzapmatrix a 4c c a mwishopmatrix anzapmatrix a3 12ac2 - 4a2 - 16c2 5a 12a2 c 16c3 - 32ac 20c 3a2 c 4c3 - 8ac 5c a3 12ac2 - 4a2 - 16c2 5a mwishopmatrixendalignTena kwa kulinganisha maingizo, tunapataanzaaligna3 12ac2 - 4a2 - 16c2 5a 10, 3a2 c 4c3 - 8ac 5c 5.mwishoalignKishaa3 12ac2 - 4a2 - 16c2 5a - 2 3a2 c 4c3 - 8ac 5c 0.Kupanua, tunapataa3 - 6a2 c 12ac2 - 8c3 - 4a2 16ac - 16c2 5a - 10c 0,ambayo tunaweza kuandika kamaa - 2c3 - 4a - 2c2 5a - 2c 0.Acha x a - 2c, hivyox3 - 4x2 5x 0, ambayo vipengele kama xx2 - 4x 5 0. Kipengele cha quadratic hakina mizizi halisi, kwa hivyo x 0, ambayo ina maana a 2c.Kubadilisha katika mlinganyo 3a2 c 4c3 - 8ac 5c 5, tunapata32c2 c 4c3 - 82c c 5c 5,ambayo hurahisisha hadi 16c3 - 16c2 5c - 5 0. Mambo haya kama c - 116c2 5 0, hivyo c 1. Inafuata kwamba a 2, b 4, na d 2, hivyomathbfM anzapmatrix 2 4 1 2 mwishopmatrix. 