Question

If matrix \(A = \left[ \begin{array}{ccc}-1 & 2025 & 2026 \\ 0 & 2 & 2027 \\ 0 & 0 & -1\end{array} \right]\), then the sum of all elements in \(\text{adj}(A^{-1})\) is equal to...

• A. \(1013\)
• B. \(2026\)
• C. \(3039\)
• D. \(6078\)

Hint:

adj(A inverse) = det(A inverse) times A, which equals A divided by det A.

Answer 1

The Correct Option is C

Step 1: Understand the concept
For an invertible matrix \(M\) we have \(\text{adj}(M) = |M|\,M^{-1}\). Taking \(M = A^{-1}\), we get \(\text{adj}(A^{-1}) = |A^{-1}|\,(A^{-1})^{-1} = \dfrac{A}{|A|}\).

Step 2: Find the determinant
\(A\) is upper triangular, so \(|A|\) is the product of the diagonal: \((-1)(2)(-1) = 2\).

Step 3: Sum the elements of A
\(-1 + 2025 + 2026 + 0 + 2 + 2027 + 0 + 0 + (-1) = 6078\).

Step 4: Result
The sum of all elements of \(\text{adj}(A^{-1}) = \dfrac{6078}{2} = 3039\). Option (C). The value 6078 is the sum of A itself, without dividing by the determinant.

Final Answer:
The sum is 3039. This is option (C). \[ \boxed{\text{(C) }3039} \]