Let A be a 3 3 non-singular matrix such that adj(A) = 2A^T . Then the value of ₂ ( |adj(adj(2A))| ) is equal to

Step-by-step solution

Given adj(A) = 2A^T . Taking determinant on both sides, |adj(A)| = |2A^T| . Using properties |adj(A)| = |A|^ n-1 and |kA| = k^n |A| for a matrix of order n=3 , we get: |A|^2 = 2^3 |A^T| . Since |A^T| = |A| , we have |A|^2 = 8|A| . As A is non-singular, |A| 0 , which gives |A| = 8 . Now, we need to find |adj(adj(2A))| . Using the property |adj(adj(X))| = |X|^ (n-1)^2 , for n=3 , we get |adj(adj(2A))| = |2A|^4 . Again using |kX| = k^n |X| , we have |2A| = 2^3 |A| = 8 8 = 64 = 2^6 . Therefore, |2A|^4 = (2^6)^4 = 2²⁴ .