Let A be a 3 3 invertible matrix. If | adj ( A 2 )| = |4 A⁻¹|^2 |A^T| , then the value of | 1 8 adj (A)| is equal to

Step-by-step solution

We know that for a matrix of order n , | adj (A)| = |A|^ n-1 and |kA| = k^n |A| . Given that A is a 3 3 matrix, we simplify the left hand side (LHS): | adj ( A 2 )| = | A 2 |^2 = (( 1 2 )^3 |A|)^2 = 1 2^6 |A|^2 = 2⁻⁶ |A|^2 Now, we simplify the right hand side (RHS): |4 A⁻¹|^2 |A^T| = (4^3 |A⁻¹|)^2 |A| = (2^6 |A|⁻¹)^2 |A| = 2¹² |A|⁻² |A| = 2¹² |A|⁻¹ Equating LHS and RHS: 2⁻⁶ |A|^2 = 2¹² |A|⁻¹ |A|^3 = 2¹⁸ |A| = 2^6 = 64 We need to find the value of | 1 8 adj (A)| : | 1 8 adj (A)| = ( 1 8 )^3 | adj (A)| = 2⁻⁹ |A|^2 Su