Animated Solution for Mathematics - Matrices and Determinants: Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$, $A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $Q=PA{P}^T$. If $P^T{Q}^{2007}P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $2a+b-3c-4d$ is equal to

When you encounter a matrix like do not panic. Your eyes should immediately recognize these as the sine and cosine values of $30^{\circ }$. This is a standard rotation matrix.

The first step in our journey is to verify its orthogonality. By calculating $P{P}^T$, we find it equals the identity matrix $I$. This property is our golden key.

Now, consider the expression $Q=PA{P}^T$. This is a similarity transformation. When we raise $Q$ to the power of $2007$, we are essentially sandwiching $A$ between $P$ and $P^T$ repeatedly.

Because $P^{T}P=I$, the inner terms collapse like a house of cards:

By induction, we can generalize this to:

Now, the final expression $P^T{Q}^{2007}P$ simplifies significantly:

Now, examine the matrix $A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$. Let us calculate the first few powers:

The pattern is undeniable. For any integer $n$:

Thus, for our specific case:

Comparing this result to the general form $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, we identify the components: $a=1,b=2007,c=0,d=1$

The final calculation $2a+b-3c-4d$ becomes:

The final answer is 2005.

This problem teaches us that in the world of JEE Advanced, brute force is rarely the answer. Instead, we look for the underlying structure—the rotation, the similarity, and the pattern. When you see a matrix power, do not reach for the calculator; reach for the pattern.