Let A be a 3 3 non-singular matrix such that |A| = 8 . If | adj (k A)| = |m A^T adj (A)| for some positive real numbers k and m , then which of the following relations is correct?

Step-by-step solution

For a matrix of order n=3 , the properties of determinants are | adj (A)| = |A|^2 and |kA| = k^3 |A| . Simplifying the left hand side (LHS): | adj (kA)| = |kA|^2 = (k^3 |A|)^2 = k^6 |A|^2 Since |A| = 8 , |A|^2 = 64 . Thus, LHS = 64 k^6 . Simplifying the right hand side (RHS): |m A^T adj (A)| = m^3 |A^T adj (A)| Using the product rule for determinants, |A^T adj (A)| = |A^T| | adj (A)| = |A| |A|^2 = |A|^3 . Thus, RHS = m^3 |A|^3 = m^3 (8^3) = 512 m^3 . Equating LHS and RHS: 64 k^6 = 512 m^3 k^6 = 8 m^3 Taking the cub