Determine the discrete convolution of the signal $s = (9 \ \ 9 \ \ 6 \ \ 9)^T$ and the convolution kernel $a = (2 \ \ 3 \ \ 1 \ \ 5)^T$. Given the convolution matrix $A \in \mathbb{R^{4 \times 4}}$, as well as the result of the convolution $\hat s = a * s$.
I'm having trouble understanding how to derive the convolution matrix. From some of the videos I have seen, it's basically the circulant matrix of the input signal:
$$A=\begin{pmatrix}9&9&6&9\\ 9&9&9&6\\ 6&9&9&9\\ 9&6&9&9\end{pmatrix}$$
However, I thought that the convolution matrix should be the circulant matrix of the convolution kernel:
$$A=\begin{pmatrix}2&5&1&3\\ 3&2&5&1\\ 1&3&2&5\\ 5&1&3&2\end{pmatrix}$$
Interestingly enough, multiplying the first matrix by $a$ and multiplying the second matrix by $s$ gives me the same result:
$$\begin{pmatrix}9&9&6&9\\ 9&9&9&6\\ 6&9&9&9\\ 9&6&9&9\end{pmatrix} * \begin{pmatrix}2\\ 3\\ 1\\ 5\end{pmatrix} = \begin{pmatrix}2&5&1&3\\ 3&2&5&1\\ 1&3&2&5\\ 5&1&3&2\end{pmatrix} * \begin{pmatrix}9\\ 9\\ 6\\ 9\end{pmatrix} = \begin{pmatrix}96\\ 84\\ 93\\ 90\end{pmatrix}$$
I just want to know which convolution matrix is correct, if any. If none of them are correct, how can I find the correct one?
Here is the problem in German (please ignore the answers in the image as they are wrong):
