GATE EC 2013 Set 1 Q39 — Linear Algebra

The $\text{DFT}$ of a vector $\begin{bmatrix} a & b & c & d \end{bmatrix}$ is the vector $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}.$ consider the product $\begin{bmatrix} p & q & r & s \end{bmatrix}=\begin{bmatrix} a & b & c & d \end{bmatrix}\begin{bmatrix} a& b&c &d \\ d&a &b &c \\c & d& a&b \\ b& c&d &a \end{bmatrix}.$

The $\text{DFT}$ of a vector $\begin{bmatrix} p & q & r & s \end{bmatrix}$ is a scaled version of