A linear state system is given with the following matrices $\boldsymbol{A}$ and $\boldsymbol{C}$. $$ \boldsymbol{A} = \begin{bmatrix} μ & 1 & 0 & \dots & 0 \\ 0 & μ & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 &\ldots & 1 \\ 0 & 0 & 0 &\ldots & μ \end{bmatrix}$$$$ \boldsymbol{C} = \begin{bmatrix} c_1 & c_2 & \dots & c_n \\ \end{bmatrix} $$ And the observability matrix $\boldsymbol{\mathcal{O}}$ is equal to
\begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix} How can we show that the rank of the observability matrix is $n$ (the state system is observable) if and only if $c_1$ is nonzero? Remark: empirically, I found that the determinant of $\boldsymbol{\mathcal{O}}$ is equal to ${c_1}^n$, however, I need analytical proof.