Let A be an n n non-singular matrix (where n > 1 ). If for all such matrices A , the identities | adj ( adj ( adj (A)))| = |A|²⁷ and |A⁻² adj ( adj (A))| = |A|^k hold true, then the value of n + k is equal to

Step-by-step solution

For an n n matrix A , the determinant of the nested adjoint is given by | adj ( adj ( adj (A)))| = |A|^ (n-1)^3 . Given that this equals |A|²⁷ , we can equate the exponents: (n-1)^3 = 27 n-1 = 3 n = 4 . Now consider the second expression: |A⁻² adj ( adj (A))| . Using the property |AB| = |A||B| , we can split this as |A⁻²| | adj ( adj (A))| . We know that |A⁻²| = |A|⁻² and | adj ( adj (A))| = |A|^ (n-1)^2 . Substituting n = 4 , we get | adj ( adj (A))| = |A|^ (4-1)^2 = |A|^9 . Thus, |A⁻² adj ( adj (A))| = |A|⁻² |A|^