If matrix A = bmatrix -1 & 2025 & 2026 0 & 2 & 2027 0 & 0 & -1 bmatrix , then the sum of all elements in adj (A⁻¹) is equal to...

Step-by-step solution

For any invertible matrix M , we have the property adj (M) = |M| M⁻¹ . Substituting M = A⁻¹ , we get: adj (A⁻¹) = |A⁻¹| (A⁻¹)⁻¹ = 1 |A| A Since A is an upper triangular matrix, its determinant is the product of its diagonal elements: |A| = (-1) 2 (-1) = 2 Thus, adj (A⁻¹) = 1 2 A . The sum of all elements in adj (A⁻¹) will be 1 2 times the sum of all elements in A . Sum of all elements in A = -1 + 2025 + 2026 + 0 + 2 + 2027 + 0 + 0 + (-1) Sum = 2025 + 2026 + 2027 = 6078 Therefore, the sum of all elements in adj (A⁻¹