# How do I detect if a zero pole cancelation has happened when looking for a zero in a MIMO transfer function?

Consider a transfer function for a MIMO control system:

$$G(s)=\begin{pmatrix} \frac{2}{s+1} & \frac{3}{s+2}\\ \frac{1}{s+1}& \frac{1}{s+1} \end{pmatrix}$$

I want to find its poles and zeros. For the poles, I know that I have to look at the roots of the pole polynomial. So, I have to look at the common divisor for the minors of order one and two, and I find that the poles are:

$$s=-1$$ (multiplicity 2)

and

$$s=-2$$ (multiplicity 1)

then I look for the zeros, by looking for the value that makes the transfer function matrix lose rank, so for the value such that:

$$det[G(s)]=0$$

and I find that there is a zero at $$s=+1$$.

By studying this topic, I have found that when looking for zeros, we need to be careful when looking for the value of $$s$$ that make the determinant equal to zero, since we could not see zero-pole cancelations, or we may miss some zeros of the system.

But, how do I know if I have missed some zero or that a zero-pole cancelation has happened?

The only way I know to find a zero is to look at the lost of rank, and I can do so by looking at the deretminant. But if a zero-pole cancelation has happened, how do I detect it?

In theory I should have a lost of controllability and/or observability if this has happened. But I am confused on how to work with zeros even if i have been trying to go deeper on this topic for days.