Let A be a 3 3 matrix such that |A| = 4 . Then the value of ₂ | adj ( 4 adj ( 2 A⁻¹ ) ) | is equal to _____ .

Step-by-step solution

Given that A is a 3 3 matrix and |A| = 4 . First, we find the determinant of A⁻¹ : |A⁻¹| = 1 |A| = 1 4 = 2⁻² Next, we evaluate the determinant of 2 A⁻¹ . Since the matrix is of order 3 3 , pulling out the scalar 2 gives: |2 A⁻¹| = 2^3 |A⁻¹| = 8 2⁻² = 2^3 2⁻² = 2^1 Now, we apply the property of the adjoint, | adj (X)| = |X|^ n-1 , where n = 3 : | adj ( 2 A⁻¹ )| = |2 A⁻¹|³⁻¹ = (2^1)^2 = 2^2 Next, we evaluate the determinant of 4 adj ( 2 A⁻¹ ) . The matrix adj ( 2 A⁻¹ ) is also of order 3 3 , so pulling out the scalar