For some , R , let A= [ array ll & 2 1 & 2 array ] and B= [ array ll 1 & 1 1 & array ] be such that A²-4 A+2 I=B²-3 B+I=O . Then ( det ( adj (A³-B³ ) ) )² is equal to _ _ _ _ .

Step-by-step solution

From A^2 - 4A + 2I = O : characteristic equation is ^2 - 4 + 2 = 0 . For A : + 2 = 4 = 2 . From B^2 - 3B + I = O : characteristic equation is ^2 - 3 + 1 = 0 . For B : + 1 = 3 = 2 . A^3 = 14A - 8I , B^3 = 8B - 3I . A^3 - B^3 = 14A - 8B - 5I = bmatrix 15 & 20 6 & 7 bmatrix . (A^3 - B^3) = 105 - 120 = -15 . For 2×2 matrix: ( adj (M)) = (M) . ( ( adj (A^3 - B^3)))^2 = 225 .