GATE EC 2015 Set 1 Q43 — Linear Algebra

Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ are related as,

$$\begin{bmatrix}A \\ B \\ C \end{bmatrix} = \begin{bmatrix}1 & 1 & 1 \\ 1 & W_3^{-1} & W_3^{-2} \\ 1 & W_3^{-2} & W_3^{-4}\end{bmatrix} \begin{bmatrix}a \\ b \\ c \end{bmatrix} \text{ where } W_3=e^{j \frac{2 \pi}{3}}$$

If another sequence $\begin{bmatrix}p, & q, & r \end{bmatrix}$ is derived as,

$$\begin{bmatrix}p \\ q \\ r \end{bmatrix} = \begin{bmatrix}1 & 1 & 1 \\ 1 & W_3^{1} & W_3^{2} \\ 1 & W_3^{2} & W_3^{4}\end{bmatrix} \begin{bmatrix}1 & 0 & 0 \\ 0 & W_3^{2} & 0 \\ 0 & 0 & W_3^{4}\end{bmatrix} \begin{bmatrix}A/3 \\ B/3 \\ C/3 \end{bmatrix} ,$$

then the relationship between the sequences $\begin{bmatrix}p, & q, & r \end{bmatrix}$ and $\begin{bmatrix}a, & b, & c \end{bmatrix}$ is