Matrices & Determinants
Orthogonal Matrix — Power of Transformed Matrix
nta_pyq_2023_apr
Grade 12
Question:
Let $P=\begin{pmatrix}\frac{\sqrt3}{2}&\frac12\\-\frac12&\frac{\sqrt3}{2}\end{pmatrix}$, $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $Q=PAP^T$. If $P^TQ^{2007}P=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, then $2a+b-3c-4d$ is equal to
Step-by-Step Solution
Key Concept: $P$ is orthogonal ($PP^T=I$). $P^TQ^{2007}P=P^T(PAP^T)^{2007}P=A^{2007}$.
$A^{2007}=\begin{pmatrix}1&2007\\0&1\end{pmatrix}$. $2a+b-3c-4d=2+2007-0-4=2005$.
Correct Answer: 2