Example of a matrix with a gersgorin disk that does not contain any eigenvalue

Question

I am looking for an example of a matrix $A\in\mathbb{C}^{n\times n}$ with the property that

1. at least one Gersgorin Disk $\Gamma_i$ contains no eigenvalue of $A$
2. for a non-empty proper subset $S$ of $N=\lbrace1,2,\dots,n\rbrace$ (i.e. $\emptyset \neq S \subsetneq N) $ it holds $$\left(\bigcup_{i\in S} \Gamma_i\right)\cap \left(\bigcup_{i\in N\setminus S} \Gamma_i\right) = \emptyset$$

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I have an example for $n=7$ (found in Gersgorin and his circles by R.S. Varga):

$A=\left(\begin{array}{ccccccc} 0&4&0&0&0&0&0 \\ 1&2&0&0&0&0&0 \\ 0&1&-2&0&0&0&0 \\ 0&0&1/8&-i&1/8&0&0 \\ 0&0&0&1/4& -2i&1/4&0 \\ 0&0&0&0&0&9/2&1/2\\ 0&0&0&0&0&1/2&-9/2 \end{array} \right)$

In this case $\Gamma_2 = \left\lbrace c\in\mathbb{C}\,:\,\left|c-2 \right|\leq 1\right\rbrace$ contains no eigenvalue of $A$, but (2) is not fulfilled for this example.

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A sketch of what I am looking for could look like this:

Here $\Gamma_2\cup\Gamma_3$ contains the eigenvalues $\lambda_2$ and $\lambda_3$, but $\Gamma_3$ contains no eigenvalue.

Do you have such an example for me or any idea how to construct such an example? ($n$ should not be to large, $3\leq n\leq5$ would be perfekt, because I want to discuss this example during a presentation)

Thanks in advance

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Due to Robert Israels answer, I have found an example with:

$A=\left(\begin{array}{ccc} 1&-1&0 \\ 2&-1&0 \\ 0&1&4 \end{array}\right)$

Now $\Gamma_1$ is empty, $\Gamma_2$ contains the eigenvalues $i$ and $-i$ and $\Gamma_3$ contains the eigenvalue $4$. Exactly what I was looking for.

Answer 1

Take an example $A$ that satisfies (1), and consider the block matrix $\pmatrix{A & 0\cr b & t\cr}$ for some suitable $b$ and $t$.