JEE Mathematics 2023 Question with Solution

Given: $P^2 = I - P$, $P\alpha + P\beta = \gamma I - 29P$, and $P\alpha - P\beta = \delta I - 13P$.

Find: $\alpha + \beta + \gamma - \delta$.

Add the two given equations:

Subtract the second equation from the first:

Using the matrix identity $P^2 = I - P$ and comparing coefficients as done in the solution, we obtain:

Now compute:

Therefore, the correct option is C.

Given: $P^2 = I - P$ and the two linear relations in $P$.

Find: $\alpha + \beta + \gamma - \delta$.

A quick way is to first add and subtract the equations to isolate $P\alpha$ and $P\beta$:

Then use the identity satisfied by $P$, namely $P^2 = I - P$, to reduce every higher power of $P$ to a linear expression in $I$ and $P$. This lets the coefficients be matched directly, giving:

Hence,

So the correct option is C.