A problem with indefinite matrix Let $A$ be a real symmmetric matrix of the type $n\times n$, positive semidefinite of the rank $k$, where $k<n$. Let 
$
K:=\left [ \begin {array}{ll}
A & b\\
b^T & d
\end{array} \right ],
$
where $d\neq 0$ and $b$ is a some matrix of the type $n \times 1$, such that $K$ is not semidefinite (is indefinite).
Then at least one principal minor of $K$ of some size is negative.
I suppose that there is a principal minor of the type $(k+1) \times (k+1)$ which is negative.
I don't know how to prove it.
 A: 
Then at least one principal minor of $K$ of some size is negative.

If $\det K$ is counted as a principal minor (technically it is), then the above conclusion holds. Sylvester's criterion says that if all principal minors of $K$ (including $\det K$ and other principal minors of smaller sizes) are nonnegative, then $K$ must be positive semidefinite. Yet, our $K$ here is supposed to indefinite. Therefore some principal minor of $K$ (possibly of size $n$ or smaller) must be negative.
However, if you mean a principal minor of size $<n$ (or of size $k+1$ in particular, as mentioned in your question), then the above statement is false in general. For a counterexample, consider
$$
K=
\left[\begin{array}{c|c}
A&b\\
\hline
b^T&d
\end{array}\right]
=\left[\begin{array}{cc|c}
1&1&0\\
1&1&1\\
\hline
0&1&1\\
\end{array}\right].
$$
Here, $K$ is indefinite (its three eigenvalues are $1$ and $1\pm\sqrt{2}$) while $A$ is a positive semidefinite matrix of rank $k=1$. However, all principal minors of $K$ of size $<n$ are nonnegative:


*

*its principal minors of size $1$ are $1,1,1$,

*its principal minors of size $2$ are $0,1,0$.

