Let A be a 4 4 real matrix such that A A^T = 2 I , where I is the identity matrix of order 4 . If |A| > 0 and B = adj (A) , then the value of ₂ | adj ( 2 B ) | is equal to _____ .

Step-by-step solution

Given the matrix equation A A^T = 2 I , we take the determinant on both sides: |A A^T| = |2 I| Using the properties |XY| = |X||Y| and |A^T| = |A| , we get: |A| |A^T| = |A|^2 For the right side, since I is a 4 4 matrix, pulling out the scalar 2 gives: |2 I| = 2^4 |I| = 16 Thus, |A|^2 = 16 . Since it is given that |A| > 0 , we have |A| = 4 . Now, we are given B = adj (A) . The determinant of B is: |B| = | adj (A)| = |A|⁴⁻¹ = |A|^3 = 4^3 = 64 = 2^6 Next, we need to find | adj ( 2 B )| . First, we find the determinant