<s>[INST] $A$ be a real symmetric matrix of size $n$ ; is $I_n+A$ always non-singular ? Is $I_n - A$ always singular ? Let $A$ be a real symmetric matrix of size $n$ ; is $I_n+A$ always non-singular ? Is $I_n - A$ always singular ?  
 [/INST] Take
$$
       I= \begin{pmatrix}
        1 & 0 \\
        0 & 1 
        \end{pmatrix},
\quad
A= \begin{pmatrix}
        0 & 1 \\
        1 & 0 
        \end{pmatrix}.
$$ We have
$$
       I+A= \begin{pmatrix}
        1 & 1 \\
        1 & 1 
        \end{pmatrix}
$$
which is clearly singular.
Take
$$
       I= \begin{pmatrix}
        1 & 0 \\
        0 & 1 
        \end{pmatrix},
\quad
A= \begin{pmatrix}
        1 & 1 \\
        1 & 1 
        \end{pmatrix}.
$$ We have
$$
       I-A= \begin{pmatrix}
        0 & -1 \\
        -1 & 0 
        \end{pmatrix}
$$
which is clearly non-singular.
</s>