JEE 2023 — Mathematics PYQ

1. Simplify the given equation

We are given the equation:

To solve for $P$, we take the transpose of both sides of equation (i):

2. Substitute $P^T$ into the new equation

Now, substitute the value of $P^T$ from equation (i) into equation (ii):

3. Solve for $P$

Rearrange the terms to group $P$:

Since we are given a > 1, we know that $a^2-1\ne 0$. Therefore, we can divide both sides by $(a^2-1)$:

4. Find the determinant and Adjoint

If $P=-I$, then the determinant of $P$ (for a $3\times 3$ matrix) is:

Now, we need to find the determinant of the adjoint of $P$. We use the property $|\text{Adj }A|=|A{|}^{n-1}$, where $n$ is the order of the matrix:

Conclusion

The value of $|\text{Adj }P|$ is 1.