# Defining the observation matrix for a Linear Kalman Filter

So I'm working on some content for an exam tomorrow and have hit a logical snag. The question can be seen below:
In particular I am referring to 5)c).

The questions asks for a measurement model to be derived for this problem.
I am aware that our measurement Zt = CtXt + noise, where Zt represents the measurement, Ct the observation matrix at time t and Xt the state vector at time t.

Xt is defined as (Pt, Vt)^T

One 'solution' I have seen for this problem is to define Ct as [1, 0]. However to me this makes little sense. In the question it states we measure both the position of the robot and the velocity. As such should this observation matrix not be [1,1]? However in which case the result of this multiplication would become = Pt + Vt. Which of course is a nonsensical measurement reading.

Instead we want an obersvation matrix defined as:

As this would give us a Zt measurement result of (Pt, Vt)^T Which the question asks for?

A similar question has been asked before, however the corresponding solution only adds to my confusion, defining an observation matrix of [0,0,1] thereby resulting in two of the three variables in the defined state vector (for that question) being multiplied by 0 and ignored?

Another source has an almost identical question, but states we measure solely the robots position and not both the position and velocity.

Question Is my thinking right, and therefore the 2x2 identity matrix for the observation matrix correct? Or is there something I am mising?

The first question you link to states that it only measures acceleration and the answer suggests to add the acceleration to the state vector as well, $$\begin{bmatrix}x^\top&\dot{x}^\top&\ddot{x}^\top\end{bmatrix}^\top$$. This is why the observation matrix becomes $$\begin{bmatrix}0&0&I\end{bmatrix}$$.
$$\begin{bmatrix} I&0\\0&I \end{bmatrix}$$
with $$I$$ a three by three identity matrix. So effectively the observation matrix would be a six by six identity matrix.