JEE Mathematics 2025 Question with Solution

Given: $A$ is a $3 \times 3$ matrix with $|A| = \frac{1}{2}$ and $\text{trace}(A) = 3$.

Find: $|B| + \text{trace}(B)$ where $B = \text{adj}(\text{adj}(2A))$.

For a square matrix of order $n = 3$, use the properties

and

Therefore,

Now,

Again using $\text{adj}(kM)=k^{n-1}\text{adj}(M)$ for $n=3$,

Also,

So,

Now compute determinant and trace:

and

Hence,

Therefore, the correct option is D.

The solution explicitly marks The Correct Option is C, but both solution approaches shown in the working compute

This matches option D, not option C = 174.

Thus, the working in the solution supports D as the defensible answer, and the listed correct-answer field is inconsistent with the actual computation.