I suspect that you mean $0 < t < 1$ in your question.
Then the answer in still no under the added condition that both matrices are not of full rank. Consider $$A_1 = \begin{pmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0\end{pmatrix}\quad 0 & -1\end{pmatrix}\quad A_2 = \begin{pmatrix}0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 1\end{pmatrix}. 0 & -1\end{pmatrix}.$$
I suspect that you mean $0 < t < 1$ in your question.
Then the answer in still no under the added condition that both matrices are not of full rank. Consider $$A_1 = \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}\quad A_2 = \begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}.$$