A question on the conditions under which $\left|\begin{smallmatrix}x+a&b&c\\a&x+b&c\\a&b&x+c\end{smallmatrix}\right|=A$ is non singular The question is if
$$
\begin{vmatrix}
x+a & b & c \\
a & x+b & c \\
a & b & x+c \\
\end{vmatrix}=A
$$
find the conditions under which it's nonsingular
on performing some ERTs
I got
$$
\begin{vmatrix}
0 & -x & 0 \\
x & x & -x \\
a+b & b & x+c \\
\end{vmatrix}=A
$$
so $$-x[x(x+c)+x(a+b))] \ne 0$$
which means
$$-x^3+x^2(a+b+c)\ne 0$$
which implies $x \ne a+b+c$
however, my book says the answer is $x \ne 0$ and $-(a+b+c)$
where am I going wrong

You made a sign error in the computation of the determinant; it should be
$-x^3 - x^2(a+b+c)$. The solutions of $-x^3 - x^2(a+b+c) = 0$ are then $x = 0$ and $x = -(a+b+c)$.
