JEE Mathematics 2025 Question with Solution

Given: $B = P A P^T$ and $C = P^T B P$, where $P$ is an orthogonal matrix, so $P^T P = I$.

Find: The sum of diagonal elements of $C$, written as $\frac{m}{n}$, and then compute $m+n$.

From

and

we get

Therefore, the sum of the diagonal elements of $C$ is the trace of $A$:

The solution states The Correct Option is B and concludes that $m+n=65$, although the intermediate working shown there is inconsistent with the matrix written in the question. Using the answer indicated by the solution, the correct option is B.

Therefore, the correct option is B.

The key property is that for the rotation matrix $P$,

So the two successive transformations undo each other when substituted into $C$.

Substitute $B = P A P^T$ into $C$:

Hence the diagonal sum of $C$ equals the diagonal sum of $A$. The provided the solution, however, explicitly marks B as the correct option. So the extracted answer is B, with the note that the displayed working on the page does not consistently match the question matrix.