UPSC NDA I 2016 Mathematics Solved Paper

Concept:The adjoint of the inverse of a matrix equals the inverse of its adjoint. This identity directly simplifies the given expression.Explanation:We need to evaluate $\text{adj}(A^{-1}) - (\text{adj }A)^{-1}$.From the property of adjoints, we know that $\text{adj}(A^{-1}) = (\text{adj }A)^{-1}$.This holds for any square matrix $A$ whose inverse exists.Substituting this identity into the expression gives:$(\text{adj }A)^{-1} - (\text{adj }A)^{-1} = 0$.The zero matrix is called the null matrix.Thus, the entire difference simplifies to the null matrix.Note: The property $\text{adj}(A^{-1}) = (\text{adj }A)^{-1}$ can be derived from $\text{adj}(A) = |A| A^{-1}$ and the fact that $|A^{-1}| = |A|^{-1}$.However, no further calculation is required because the identity directly cancels the two terms.Answer:Null matrix (Option B).