Let A be a 3 3 non-singular matrix with real entries such that |(3A) adj (3A)| = 3^9 2^3 . Then the value of | adj (2A⁻¹)| is equal to

Step-by-step solution

Given that A is a 3 3 matrix, we know that for any matrix X of order n , X adj (X) = |X|I . Taking the determinant on both sides, |X adj (X)| = ||X|I| = |X|^n . Here, X = 3A and n = 3 , so |(3A) adj (3A)| = |3A|^3 . Using the property |kA| = k^n|A| , we have |3A| = 3^3|A| = 27|A| . Therefore, |(3A) adj (3A)| = (27|A|)^3 = 3^9 |A|^3 . Equating this to the given value: 3^9 |A|^3 = 3^9 2^3 |A|^3 = 2^3 |A| = 2 . Now, we need to find | adj (2A⁻¹)| . Using the property | adj (X)| = |X|^ n-1 , we get | adj (2A⁻¹)| = |2A⁻¹