Equal number of poles and zeros for square transfer matrix

Question

In page 10 of this document (MIT Courseware on control) it is stated that since the transfer function is square, there is an equal number of poles and zeros. Does this hold and if so under which circumstances?

For example for the following system:

\begin{equation*} G(s) = \frac{1}{(s+2)} \left[\begin{array}{cc} s+1 & \frac{(s+1)}{(s+3)} \\ s+2 & \frac{(s+2)}{(s+5)}\end{array}\right] \end{equation*}

this does not hold as matlab can confirm:

>> G = 1/(s+2) * [s+1, (s+1)/(s+3); s+2, (s+2)/(s+5)];
>> pole(G)

ans =

   -2.0000
   -3.0000
   -2.0000
   -5.0000
   -2.0000

>> tzero(G)

ans =

   -1.0000
   -2.0000
   -2.0000

Answer 1

The correct form of that statement is that for an invertible transfer matrix the number of poles is equal to the number of zeroes.