= Solution
Apply the <elementary row operation> $R_1\gets R_1-R_2$, which preserves the <determinant>. Expanding along the new first row gives
$$
\begin{aligned}
\det A&=\det\begin{pmatrix}2-c&0&0\\c&c&c\\8&7&c\end{pmatrix}\\
&=(2-c)\det\begin{pmatrix}c&c\\7&c\end{pmatrix}=(2-c)c(c-7).
\end{aligned}
$$
A square <matrix> is an <invertible matrix> exactly when its <determinant> is nonzero. Therefore
$$
\boxed{c=0,\quad c=2,\quad c=7}
$$
are precisely the three values making $A$ a <singular matrix>. They also have direct explanations: at $c=0$ the second row is zero, at $c=2$ the first two rows coincide, and at $c=7$ the last two columns coincide.
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