= Gershgorin exclusion for a matrix-pencil parameter
{c}
To exclude zero as an <eigenvalue> of $A+sB$, it is enough that every row has positive real diagonal entry exceeding the sum of the absolute values of its off-diagonal entries. All <Gershgorin discs> then lie strictly in the right half-plane. For
$$
A=\begin{pmatrix}2&-2&0\\-3&5&0\\1&2&3\end{pmatrix},\qquad B=\begin{pmatrix}2&0&0\\0&2&1\\0&1&2\end{pmatrix},
$$
the three diagonal-dominance margins of $A+sB$ are $2s,2+s,s$, all positive for $s>0$. Thus $A-\mu B$ is nonsingular for negative real $\mu$. Working with the pencil directly avoids computing either a <determinant> <polynomial> or $B^{-1}A$.
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