Question on path connectedness of two invertible matrix.

Question

Suppose that $A_1 , A_2 \in M_n (\mathbb C)$ are invertible matrices.
Claim: There exists finitely many complex numbers $z\in \mathbb C$ such that $f(t) :=\det(zA_1+ (1-z)A_2)=0.$ Which is true via Fundamental theorem of algebra. Now define $$\gamma:[0,1]\longrightarrow \mathbb C.$$
Would it easily follow by the construction above that there exists a continuous path $\varphi(z)$ of the form $$\varphi(z)= \gamma(z) A_1 +(1-\gamma(z))A_2$$ connecting $A_1$ to $A_2$?

Any help and hints would be so much appreciated!