$\textbf{A}$ is a $N \times N$ matrix and $\textbf{B}$ is a $N \times M$ matrix. The polynomial degree of greatest common divisor of the largest minors of matrix $[\textbf{A}-\lambda \textbf{I} \quad \textbf{B}]$ is $0$, where $\lambda$ is the eigenvalues of $\textbf{A}$ and $\textbf{I}$ is the identity matrix. If add one column and one row to matrix $\textbf{A}$ to get the matrix $[\textbf{A}_1-\lambda \textbf{I} \quad \textbf{B}_1]$, how to prove that the polynomial degree of greatest common divisor of the largest minors of matrix $[\textbf{A}_1-\lambda \textbf{I} \quad \textbf{B}_1]$ smaller than 2?
For example, $$ \textbf{A} = \left( \matrix{0 & 0\cr a_{21} & 0} \right) $$
$$ \textbf{B} = \left( \matrix{b_1 \cr 0} \right) $$ Add one column and row to $\textbf{A}$ to get $[\textbf{A}_1-\lambda \textbf{I} \quad \textbf{B}_1]$ as $$ \left( \matrix{-\lambda & 0 & a_{13} & b_1\cr a_{21} & -\lambda & a_{23} & 0 \cr a_{31} & a_{32} & a_{33}-\lambda & b_3} \right) $$