  The variety $ \mathrm{Sing}_{n, m} $ consists of all tuples $ X =
(X_1,\ldots, X_m) $ of $ n\times n $ matrices such that every linear
combination of $ X_1,\ldots, X_m $ is singular. Equivalently,
$X\in\mathrm{Sing}_{n,m}$ if and only if $\det(\lambda_1 X_1 + \ldots +
\lambda_m X_m) = 0$ for all $ \lambda_1,\ldots, \lambda_m\in \mathbb Q $. Makam
and Wigderson asked whether the ideal generated by these equations is always
radical, that is, if any polynomial identity that is valid on $
\mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials
$\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. We answer this question in the
negative by determining the vanishing ideal of $ \mathrm{Sing}_{2, m} $ for all
$ m\in \mathbb N $. Our results exhibit that there are additional equations
arising from the tensor structure of $ X $. More generally, for any $ n $ and $
m\ge n^2 - n + 1 $, we prove there are equations vanishing on $
\mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of
type $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. Our methods are based on
classical results about Fano schemes, representation theory and Gr\"obner
bases.
