Let M and N be two 3 3 matrices such that M^2 N = 2I and |adj(adj(2M)) N⁻¹| = 32768 . Then the value of |M|^2 + |N| is equal to

Step-by-step solution

Given M^2 N = 2I . Taking determinant on both sides, |M^2 N| = |2I| . |M|^2 |N| = 2^3 |I| = 8 ... (i) Also given, |adj(adj(2M)) N⁻¹| = 32768 . Using properties |AB| = |A||B| and |A⁻¹| = 1 |A| , we get |adj(adj(2M))| |N| = 32768 . Using |adj(adj(A))| = |A|^ (n-1)^2 , for n=3 , |adj(adj(2M))| = |2M|^4 . Using |kA| = k^n |A| , |2M| = 2^3 |M| = 8|M| . So, |2M|^4 = (8|M|)^4 = 4096|M|^4 . Therefore, 4096|M|^4 |N| = 32768 |M|^4 |N| = 8 ... (ii) Multiplying (i) and (ii): (|M|^2 |N| ) ( |M|^4 |N| ) = 8 8 . |M|^6 = 64 |M|^2