Nice invertible matrices

It can be frustrating to find invertible matrices. Thus, this page introduces a plethora of matrices that are invertible with a few other "nice" conditions:

1. While finding the inverse using Gaussian elimination with partial pivoting, all calculations require usually one digit and at most two after the decimal place.
2. The matrices and the inverses have values less than $10$ in absolute value.
3. The eigenvalues are "nice", meaning that they have at most one digit after the decimal place, although this is not possible for symmetric matrices, so a weaker "niceness" is used.

The requirement that the eigenvalues are "nice" is simply a mechanism to choose from the vast number of possible examples available.

Constructing variations

1. $2 \times 2$ non-symmetric matrices with real eigenvalues
2. $2 \times 2$ non-symmetric matrices with complex eigenvalues
3. $2 \times 2$ symmetric matrices
4. $3 \times 3$ non-symmetric matrices with real eigenvalues
5. $3 \times 3$ non-symmetric matrices with complex eigenvalues
6. $3 \times 3$ symmetric matrices

$2\times2$ non-symmetric matrices with real eigenvalues

$\left(\begin{array}{rr} 2 & 8 \\ 1.6 & 8.4 \end{array}\right)$ and $\left(\begin{array}{rr} 2.1 & -2 \\ -0.4 & 0.5 \end{array}\right)$ with the first matrix having eigenvalues $10$ and $0.4$.

$\left(\begin{array}{rr} 0.5 & 1.5 \\ 0.1 & 0.7 \end{array}\right)$ and $\left(\begin{array}{rr} 3.5 & -7.5 \\ -0.5 & 2.5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $0.2$.

$\left(\begin{array}{rr} 2 & 2 \\ -0.8 & -0.6 \end{array}\right)$ and $\left(\begin{array}{rr} -1.5 & -5 \\ 2 & 5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $0.4$.

$\left(\begin{array}{rr} 2 & 3 \\ -0.8 & -1.4 \end{array}\right)$ and $\left(\begin{array}{rr} 3.5 & 7.5 \\ -2 & -5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $-0.4$.

$\left(\begin{array}{rr} 1 & 1 \\ 0.5 & 1.5 \end{array}\right)$ and $\left(\begin{array}{rr} 1.5 & -1 \\ -0.5 & 1 \end{array}\right)$ with the first matrix having eigenvalues $2$ and $0.5$.

$\left(\begin{array}{rr} 2 & 2 \\ -1.2 & -1.4 \end{array}\right)$ and $\left(\begin{array}{rr} 3.5 & 5 \\ -3 & -5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $-0.4$.

$\left(\begin{array}{rr} 0.5 & 1.5 \\ -0.3 & -1.3 \end{array}\right)$ and $\left(\begin{array}{rr} 6.5 & 7.5 \\ -1.5 & -2.5 \end{array}\right)$ with the first matrix having eigenvalues $-1$ and $0.2$.

$\left(\begin{array}{rr} 0.5 & 0.5 \\ 0.3 & 0.7 \end{array}\right)$ and $\left(\begin{array}{rr} 3.5 & -2.5 \\ -1.5 & 2.5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $0.2$.

$\left(\begin{array}{rr} 2 & 6 \\ -1.6 & -4.4 \end{array}\right)$ and $\left(\begin{array}{rr} 5.5 & 7.5 \\ 2 & 2.5 \end{array}\right)$ with the first matrix having eigenvalues $-2$ and $-0.4$.

$\left(\begin{array}{rr} 2 & 3 \\ -1.6 & -2.6 \end{array}\right)$ and $\left(\begin{array}{rr} 6.5 & 7.5 \\ -4 & -5 \end{array}\right)$ with the first matrix having eigenvalues $-1$ and $0.4$.

$\left(\begin{array}{rr} 2 & 4 \\ -1.6 & -3.6 \end{array}\right)$ and $\left(\begin{array}{rr} 4.5 & 5 \\ -2 & -2.5 \end{array}\right)$ with the first matrix having eigenvalues $-2$ and $0.4$.

$\left(\begin{array}{rr} 0.5 & 2 \\ 0.4 & 2.1 \end{array}\right)$ and $\left(\begin{array}{rr} 8.4 & -8 \\ -1.6 & 2 \end{array}\right)$ with the first matrix having eigenvalues $2.5$ and $0.1$.

$\left(\begin{array}{rr} 0.5 & 1.5 \\ 0.4 & 1.6 \end{array}\right)$ and $\left(\begin{array}{rr} 8 & -7.5 \\ -2 & 2.5 \end{array}\right)$ with the first matrix having eigenvalues $2$ and $0.1$.

$\left(\begin{array}{rr} 0.5 & 0.5 \\ 0.4 & 0.6 \end{array}\right)$ and $\left(\begin{array}{rr} 6 & -5 \\ -4 & 5 \end{array}\right)$ with the first matrix having eigenvalues $1$ and $0.1$.

$2\times2$ non-symmetric matrices with complex eigenvalues

$\left(\begin{array}{rr} 0.5 & 0.5 \\ -0.4 & 0.1 \end{array} \right)$ and $\left(\begin{array}{rr} 0.4 & -2 \\ 1.6 & 2 \end{array} \right)$ with the first matrix having eigenvalues $0.3 \pm 0.4j$.

$\left(\begin{array}{rr} 2 & 8 \\ -0.4 & 0.4 \end{array} \right)$ and $\left(\begin{array}{rr} 0.1 & -2 \\ 0.1 & 0.5 \end{array} \right)$ with the first matrix having eigenvalues $1.2 \pm 1.6j$.

$\left(\begin{array}{rr} 2 & 4 \\ -0.4 & -0.4 \end{array} \right)$ and $\left(\begin{array}{rr} -0.5 & -5 \\ 0.5 & 2.5 \end{array} \right)$ with the first matrix having eigenvalues $0.8 \pm 0.4j$.

$\left(\begin{array}{rr} 0.5 & 2 \\ -0.1 & 0.1 \end{array} \right)$ and $\left(\begin{array}{rr} 0.4 & -8 \\ 0.4 & 2 \end{array} \right)$ with the first matrix having eigenvalues $0.3 \pm 0.4j$.

$\left(\begin{array}{rr} 2 & 2 \\ -0.8 & 1.2 \end{array} \right)$ and $\left(\begin{array}{rr} 0.3 & -0.5 \\ 0.2 & 0.5 \end{array} \right)$ with the first matrix having eigenvalues $1.6 \pm 1.2j$.

$\left(\begin{array}{rr} 0.5 & 0.5 \\ -0.2 & 0.3 \end{array} \right)$ and $\left(\begin{array}{rr} 1.2 & -2 \\ 0.8 & 2 \end{array} \right)$ with the first matrix having eigenvalues $0.4 \pm 0.3j$.

$\left(\begin{array}{rr} 2 & 3 \\ -1.2 & -1.6 \end{array} \right)$ and $\left(\begin{array}{rr} -4 & -7.5 \\ 3 & 5 \end{array} \right)$ with the first matrix having eigenvalues $0.2 \pm 0.6j$.

$\left(\begin{array}{rr} 0.5 & 2 \\ -0.4 & -1.1 \end{array} \right)$ and $\left(\begin{array}{rr} -4.4 & -8 \\ 1.6 & 2 \end{array} \right)$ with the first matrix having eigenvalues $-0.3 \pm 0.4j$.

$\left(\begin{array}{rr} 2 & 4 \\ -1.6 & -2.8 \end{array} \right)$ and $\left(\begin{array}{rr} -3.5 & -5 \\ 2 & 2.5 \end{array} \right)$ with the first matrix having eigenvalues $-0.4 \pm 0.8j$.

$\left(\begin{array}{rr} 2 & 8 \\ -1.6 & -4.4 \end{array} \right)$ and $\left(\begin{array}{rr} -1.1 & -2 \\ 0.4 & 0.5 \end{array} \right)$ with the first matrix having eigenvalues $-1.2 \pm 1.6j$.

$\left(\begin{array}{rr} 2 & 2 \\ -1.6 & 0.4 \end{array} \right)$ and $\left(\begin{array}{rr} 0.1 & -0.5 \\ 0.4 & 0.5 \end{array} \right)$ with the first matrix having eigenvalues $1.2 \pm 1.6j$.

$\left(\begin{array}{rr} 2 & 2 \\ -1.8 & -1.6 \end{array} \right)$ and $\left(\begin{array}{rr} -4 & -5 \\ 4.5 & 5 \end{array} \right)$ with the first matrix having eigenvalues $0.2 \pm 0.6j$.

$2\times2$ symmetric matrices

None of these have "nice" eigenvalues and all have determinant equal to $1$. You can negate both the off diagonal entries, or multiply one matrix by $\pm0.1, \pm0.2, \pm0.4, \pm0.5, \pm2, \pm2.5, \pm5$ or $\pm10$ and then multiply the other by the reciprocal, and the arithmetic will still be "nice".

$\left(\begin{array}{rr} 5 & 1 \\ 1 & 0.4 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 0.4 & -1 \\ -1 & 5 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 2.5 & 0.5 \\ 0.5 & 0.5 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 0.5 & -0.5 \\ -0.5 & 2.5 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 5 & 2 \\ 2 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 1 & -2 \\ -2 & 5 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 2.5 & 1 \\ 1 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 0.8 & -1 \\ -1 & 2.5 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 0.4 & 0.2 \\ 0.2 & 2.6 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 2.6 & -0.2 \\ -0.2 & 0.4 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 2 & 1 \\ 1 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 1 & -1 \\ -1 & 2 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 5 & 4 \\ 4 & 3.4 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 3.4 & -4 \\ -4 & 5 \\ \end{array}\right)$.

$\left(\begin{array}{rr} 2.5 & 2 \\ 2 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rr} 2 & -2 \\ -2 & 2.5 \\ \end{array}\right)$.

$3\times3$ non-symmetric matrices with real eigenvalues

It seems there are very few "nice" invertible $3 \times 3$ matrices that have only one digit beyond the decimal point, so these appear first followed by others that have no more than four entries with two digits beyond the decimal point Those with the fewest appear first.

$\left(\begin{array}{rrr} 5 & 9 & 2 \\ -3 & -6.4 & -2.2 \\ 3.5 & 8.8 & 4.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.4 & 5.3 & 1.4 \\ -1.4 & -3.5 & -1 \\ 0.8 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $2$ and $-1$.

$\left(\begin{array}{rrr} 2 & -2 & 3.2 \\ 1 & 0.5 & -0.7 \\ -1 & 3.5 & -6.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.3 & 0.5 & 0.1 \\ -3.4 & 4.5 & -2.3 \\ -2 & 2.5 & -1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $1$ and $0.4$.

$\left(\begin{array}{rrr} 2 & 0 & -4 \\ 0.6 & -0.4 & -1 \\ -1 & 1 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.3 & 2 & 0.8 \\ 0.7 & -2 & 0.2 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $1$ and $-0.4$.

$\left(\begin{array}{rrr} 2 & -2 & 6 \\ 0.6 & -0.8 & 3.6 \\ -1 & 1.5 & -5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.4 & 1 & 2.4 \\ 0.6 & 4 & 3.6 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $1$ and $0.2$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ 0.3 & -0.2 & -0.5 \\ -0.5 & 0.5 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 4 & 1.6 \\ 1.4 & -4 & 0.4 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $0.5$ and $-0.2$.

$\left(\begin{array}{rrr} 1 & -3 & 3 \\ -0.1 & 0.9 & -0.7 \\ -0.5 & 2.5 & -5.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.6 & 4.5 & 0.3 \\ 0.1 & 2 & -0.2 \\ -0.1 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $1$ and $0.4$.

$\left(\begin{array}{rrr} 1 & -3 & -0.8 \\ -0.1 & 0.9 & 0.6 \\ -0.5 & 2.5 & 0.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.4 & 0.5 & 2.7 \\ 0.6 & -0.5 & 1.3 \\ -0.5 & 2.5 & -1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-0.4$ and $0.4$.

$\left(\begin{array}{rrr} 1 & -3 & 3 \\ -0.6 & 2.1 & -2 \\ -0.5 & 2 & -3.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6.7 & 9 & 0.6 \\ 2.2 & 4 & -0.4 \\ 0.3 & 1 & -0.6 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.1$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ -0.3 & -1 & 0.2 \\ -0.1 & 2.5 & 2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 5.4 & 5 & 2 \\ -1.28 & -2 & -0.4 \\ 1.7 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-1$ and $0.2$.

$\left(\begin{array}{rrr} 2 & 1.2 & -4 \\ -1.2 & -1 & 2 \\ -1 & -0.2 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 5 & 2 \\ -3.5 & -5 & -1 \\ 0.95 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $0.4$ and $-0.4$.

$\left(\begin{array}{rrr} 1 & 0 & -0.2 \\ 0.3 & -2 & -1.46 \\ -0.5 & 5 & 4.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.9 & 1 & 0.4 \\ 0.5 & -4 & -1.4 \\ -0.5 & 5 & 2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $1$ and $-0.4$.

$\left(\begin{array}{rrr} 0.2 & 0 & -1 \\ 0.06 & -0.4 & -0.1 \\ -0.1 & 1 & 2.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 5 & 2 \\ 0.7 & -2 & 0.2 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-0.4$ and $0.2$.

$\left(\begin{array}{rrr} 5 & 3 & -1 \\ -4.5 & -3.7 & -0.1 \\ -1 & 1.9 & 3.7 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.7 & 2.6 & 0.8 \\ -3.35 & -3.5 & -1 \\ 2.45 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-1$ and $1$.

$\left(\begin{array}{rrr} 5 & 9 & -1 \\ -4.5 & -8.5 & 0.5 \\ -1 & -0.8 & 2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 9.15 & 9.5 & 2 \\ -4.7 & -5 & -1 \\ 2.45 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-4$, $2.5$ and $0.2$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ -0.45 & -0.4 & 0.5 \\ -0.1 & 1 & 2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6.9 & 5 & 2 \\ -4.7 & -5 & -1 \\ 2.45 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-0.4$ and $0.2$.

$\left(\begin{array}{rrr} 5 & 6 & -4 \\ -4.5 & -6.1 & 3.2 \\ -2.5 & -2 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3.6 & 3.2 & 1.04 \\ -2 & -2 & -0.4 \\ 1.25 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-2.5$ and $0.4$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ 0.15 & -1 & -1.3 \\ -0.25 & 2.5 & 5.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.8 & 2 & 0.8 \\ 0.4 & -2 & -0.4 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-0.5$ and $0.5$.

$\left(\begin{array}{rrr} 0.2 & 0 & -0.4 \\ 0.06 & -0.4 & -1.12 \\ -0.1 & 1 & 5.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.8 & 2 & 0.8 \\ 1 & -5 & -1 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-0.2$ and $0.2$.

$\left(\begin{array}{rrr} 0.5 & -1.5 & -1 \\ -0.05 & 0.4 & 0.35 \\ 0.2 & 1.9 & 0.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.5 & 7 & 0.5 \\ -0.3 & -1 & 0.5 \\ 0.7 & 5 & -0.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1$, $-0.5$ and $0.5$.

$\left(\begin{array}{rrr} 2.5 & 0 & -5 \\ -1.5 & -0.2 & 3.12 \\ 0.5 & 0.5 & -0.3 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3 & 5 & 2 \\ -2.22 & -3.5 & 0.6 \\ 1.3 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.5$.

$\left(\begin{array}{rrr} 2 & -6 & 6 \\ -1.2 & 4.2 & -4 \\ -1 & 4 & -7 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3.35 & 4.5 & 0.3 \\ 1.1 & 2 & -0.2 \\ 0.15 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $4$ and $0.2$.

$\left(\begin{array}{rrr} 2.5 & -4 & 1 \\ -1.5 & 2.1 & 1.6 \\ -1.25 & 3 & -4.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 5.7 & 6 & 3.4 \\ 3.5 & 4 & 2.2 \\ 0.75 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $5$ and $0.1$.

$\left(\begin{array}{rrr} 1 & 0 & -0.8 \\ -0.6 & -0.6 & 0.56 \\ -0.5 & 2 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4 & 4 & 1.2 \\ -0.5 & -1 & 0.2 \\ 3.75 & 5 & 1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-1$ and $0.2$.

$\left(\begin{array}{rrr} 0.5 & 2 & 1.5 \\ -0.3 & -2.6 & -2.7 \\ -0.4 & 0.4 & 2.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6.2 & 5 & 1.5 \\ -1.92 & -2 & -0.9 \\ 1.16 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-2$ and $0.2$.

$\left(\begin{array}{rrr} 2 & -2 & -0.4 \\ 1.6 & -1.8 & 1.04 \\ -1 & 1.5 & -0.7 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.3 & 2 & 2.8 \\ -0.08 & 1.8 & 2.72 \\ -0.6 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$, $1$ and $0.5$.

$\left(\begin{array}{rrr} 2 & -2 & -2.8 \\ 1 & -1.2 & -0.28 \\ -1 & 1.5 & 1.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.9 & 2 & 2.8 \\ 0.82 & 0.6 & 2.24 \\ -0.3 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-1$ and $0.4$.

$\left(\begin{array}{rrr} 2 & -2 & -1.6 \\ 1 & -1.7 & 1.04 \\ -1 & 2 & -0.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.7 & 2 & 2.4 \\ 0.32 & 1.2 & 1.84 \\ -0.15 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.4$.

$\left(\begin{array}{rrr} 2.5 & 0 & -2 \\ -1.5 & -1.5 & 1.4 \\ -1.25 & 5 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.6 & 1.6 & 0.48 \\ -0.2 & -0.4 & 0.08 \\ 1.5 & 2 & 0.6 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-2.5$ and $0.5$.

$\left(\begin{array}{rrr} 0.2 & 0 & -1 \\ 0.06 & -0.4 & -0.1 \\ 0.14 & 1 & 1.3 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.1 & 5 & 2 \\ 0.46 & -2 & 0.2 \\ -0.58 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1$, $0.5$ and $-0.4$.

$\left(\begin{array}{rrr} 0.4 & 0 & -2 \\ 0.12 & -1.4 & -2.4 \\ 0.08 & 2 & 3.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.3 & 5 & 3.5 \\ 0.78 & -2 & -0.9 \\ -0.44 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $1$ and $-0.4$.

$\left(\begin{array}{rrr} 1 & 0 & -5 \\ 0.3 & -0.2 & 0.1 \\ -0.2 & 0.5 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.45 & 2.5 & 1 \\ 0.62 & -1 & 1.6 \\ -0.11 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $0.8$ and $-0.5$.

$\left(\begin{array}{rrr} 1 & 1.2 & -0.2 \\ 0.3 & -3.14 & -0.56 \\ -0.2 & 4.76 & 1.04 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 2.2 & 1.3 \\ 0.2 & -1 & -0.5 \\ -0.8 & 5 & 3.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $1$ and $0.4$.

$\left(\begin{array}{rrr} 1 & 0 & -5 \\ 0.3 & -2 & 0.1 \\ -0.5 & 5 & 3.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.75 & 2.5 & 1 \\ 0.11 & -0.1 & 0.16 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $4$, $-2.5$ and $1$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ 0.3 & -2 & -0.2 \\ -0.5 & 5 & 5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.9 & 1 & 0.4 \\ 0.14 & -0.4 & 0.04 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-2$ and $1$.

$\left(\begin{array}{rrr} 1 & 0 & -0.8 \\ 0.3 & -2 & 1.36 \\ -0.5 & 5 & -2.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 2 & 0.8 \\ -0.05 & 1.5 & 0.8 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $1$ and $0.4$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ 0.15 & -1 & -0.1 \\ -0.25 & 2.5 & 2.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.8 & 2 & 0.8 \\ 0.28 & -0.8 & 0.08 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-1$ and $0.5$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ 0.15 & -1 & -0.1 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.5 & 5 & 2 \\ 0.25 & -0.5 & 0.2 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1$, $-1$ and $0.5$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ 0.15 & -1 & -0.7 \\ -0.25 & 2.5 & 2.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.5 & 5 & 2 \\ 0.4 & -2 & -0.4 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.5$.

$\left(\begin{array}{rrr} 0.5 & 0 & -0.4 \\ 0.15 & -1 & 0.68 \\ -0.25 & 2.5 & -1.3 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.6 & 4 & 1.6 \\ -0.1 & 3 & 1.6 \\ -0.5 & 5 & 2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $0.5$ and $0.2$.

$\left(\begin{array}{rrr} 0.4 & 0 & -0.8 \\ 0.12 & -0.8 & 1.6 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.4 & 1 & 0.4 \\ 0.26 & -0.1 & 0.46 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$, $2$ and $0.4$.

$\left(\begin{array}{rrr} 0.4 & 0 & -2 \\ 0.12 & -0.8 & -0.2 \\ -0.2 & 2 & 5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.25 & 2.5 & 1 \\ 0.35 & -1 & 0.1 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-0.8$ and $0.4$.

$\left(\begin{array}{rrr} 0.4 & 0 & -2 \\ 0.12 & -0.8 & 0.88 \\ -0.2 & 2 & -0.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2 & 5 & 2 \\ 0.19 & 0.6 & 0.74 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$, $1$ and $0.4$.

$\left(\begin{array}{rrr} 0.4 & 0 & -0.8 \\ 0.12 & -0.8 & 0.1 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.1 & 4 & 1.6 \\ 0.29 & -0.4 & 0.34 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1$, $-1$ and $0.4$.

$\left(\begin{array}{rrr} 0.2 & 0 & -0.4 \\ 0.06 & -0.4 & 0.8 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.8 & 2 & 0.8 \\ 0.52 & -0.2 & 0.92 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-1$, $1$ and $0.2$.

$\left(\begin{array}{rrr} 0.2 & 0 & -1 \\ 0.06 & -0.4 & 0.14 \\ -0.1 & 1 & 1.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 5 & 2 \\ 0.64 & -1.4 & 0.44 \\ -0.1 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.2$.

$\left(\begin{array}{rrr} 0.2 & 0 & -0.4 \\ 0.06 & -0.4 & -0.04 \\ -0.1 & 1 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 5 & 2 \\ 0.7 & -2 & 0.2 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1$, $-0.4$ and $0.2$.

$\left(\begin{array}{rrr} 0.2 & 0 & -0.4 \\ 0.06 & -0.4 & -0.52 \\ -0.1 & 1 & 2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 5 & 2 \\ 1 & -5 & -1 \\ -0.25 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $0.2$ and $-0.2$.

$\left(\begin{array}{rrr} 0.1 & 0 & -0.2 \\ 0.03 & -0.2 & 2.74 \\ -0.05 & 0.5 & -1.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 9.9 & 1 & 0.4 \\ 0.8 & 2 & 2.8 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $0.4$ and $0.1$.

$\left(\begin{array}{rrr} 0.1 & 0 & -0.2 \\ 0.03 & -0.2 & 0.4 \\ -0.05 & 0.5 & 0.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 9.6 & 4 & 1.6 \\ 1.04 & -0.4 & 1.84 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-0.5$, $0.5$ and $0.1$.

$\left(\begin{array}{rrr} 0.2 & 0 & -1 \\ -0.02 & -0.3 & 0.5 \\ 0.04 & 1 & 1.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 5.2 & 5 & 1.5 \\ -0.28 & -2 & 0.4 \\ 0.04 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.2$.

$\left(\begin{array}{rrr} 2 & 2.4 & 0.8 \\ -0.2 & -1.74 & -1.08 \\ -0.4 & 4.52 & 3.84 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.9 & 2.8 & 0.6 \\ -0.6 & -4 & -1 \\ 0.8 & 5 & 1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $2$ and $-0.4$.

$\left(\begin{array}{rrr} 1 & -3 & 1.6 \\ -0.1 & 0.9 & 1.12 \\ -0.5 & 2.5 & -2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.3 & 1 & 2.4 \\ 0.38 & 0.6 & 0.64 \\ -0.1 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.4$.

$\left(\begin{array}{rrr} 1 & -3 & 0.4 \\ -0.1 & 0.9 & 0.88 \\ -0.5 & 2.5 & -2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2 & 2.5 & 1.5 \\ 0.32 & 0.9 & 0.46 \\ -0.1 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.4$.

$\left(\begin{array}{rrr} 5 & -5 & -7 \\ -0.5 & 0.7 & 1.06 \\ -2.5 & 4.5 & 5.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 3 & 0.2 \\ 0.05 & -4 & 0.9 \\ 0.25 & 5 & -0.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $10$, $1$ and $-0.2$.

$\left(\begin{array}{rrr} 5 & -5 & -7 \\ -0.5 & -1 & 1.4 \\ -2.5 & 7.5 & 4.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 1.2 & 0.56 \\ 0.05 & -0.2 & 0.14 \\ 0.25 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $10$, $-2.5$ and $1$.

$\left(\begin{array}{rrr} 1 & -1 & -1.4 \\ -0.1 & -0.2 & 0.28 \\ -0.5 & 1.5 & 0.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3 & 6 & 2.8 \\ 0.25 & -1 & 0.7 \\ 1.25 & 5 & 1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.2$.

$\left(\begin{array}{rrr} 2 & -2 & -4 \\ -0.2 & -1.8 & 2 \\ -1 & 6 & -1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.55 & 6.5 & 2.8 \\ 0.55 & 1.5 & 0.8 \\ 0.75 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $4$ and $0.2$.

$\left(\begin{array}{rrr} 1 & -3 & 1.6 \\ -0.1 & -1.7 & 4.24 \\ -0.5 & 6.5 & -6.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.6 & 1 & 1 \\ 0.28 & 0.6 & 0.44 \\ 0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-10$, $2$ and $0.5$.

$\left(\begin{array}{rrr} 1 & -1 & -0.8 \\ -0.1 & -0.9 & 3.28 \\ -0.5 & 3 & -2.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.5 & 1 & 0.8 \\ 0.38 & 0.6 & 0.64 \\ 0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$, $2$ and $0.5$.

$\left(\begin{array}{rrr} 1 & -1 & -0.8 \\ -0.1 & -0.3 & 2.56 \\ -0.5 & 1.5 & -0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.8 & 1 & 1.4 \\ 0.68 & 0.6 & 1.24 \\ 0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.4$.

$\left(\begin{array}{rrr} 1 & -3 & 1.6 \\ -0.1 & 0.1 & 2.08 \\ -0.5 & 2 & -1.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.3 & 1 & 6.4 \\ 1.18 & 0.6 & 2.24 \\ 0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$, $2$ and $0.2$.

$\left(\begin{array}{rrr} 1 & -1 & -5 \\ -0.6 & -0.1 & 2 \\ 0.2 & 0.8 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.25 & 7.5 & 6.25 \\ -2.5 & -5 & -2.5 \\ 1.15 & 2.5 & 1.75 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-0.5$ and $0.4$.

$\left(\begin{array}{rrr} 2 & -2 & -1.6 \\ -1.2 & -0.3 & 1.16 \\ -1 & 6 & 1.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.5 & 1.2 & 0.56 \\ -0.2 & -0.4 & 0.08 \\ 1.5 & 2 & 0.6 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-2$ and $0.5$.

$\left(\begin{array}{rrr} 2 & 0 & -0.4 \\ -1.2 & -0.3 & 1 \\ -1 & 1 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.65 & 0.2 & 0.06 \\ -0.1 & -0.8 & 0.76 \\ 0.75 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $1$ and $-0.8$.

$\left(\begin{array}{rrr} 1 & 0 & -1.4 \\ -0.6 & -0.6 & 0.14 \\ -0.5 & 2 & 4.7 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3.1 & 2.8 & 0.84 \\ -2.75 & -4 & -0.7 \\ 1.5 & 2 & 0.6 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $0.5$ and $-0.4$.

$\left(\begin{array}{rrr} 0.5 & -0.5 & -1 \\ -0.3 & -0.3 & 0.68 \\ -0.25 & 2.25 & 0.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 9 & 9 & 3.2 \\ -0.5 & -1 & 0.2 \\ 3.75 & 5 & 1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2$, $-1$ and $0.1$.

$\left(\begin{array}{rrr} 0.5 & 0 & -0.4 \\ -0.3 & -0.3 & 0.64 \\ -0.25 & 1 & 2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.6 & 0.8 & 0.24 \\ -1 & -2 & 0.4 \\ 0.75 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-0.5$ and $0.4$.

$\left(\begin{array}{rrr} 2 & 0 & -4 \\ -1.2 & -0.4 & 3.08 \\ -1 & 1 & 2.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.1 & 2 & 0.8 \\ -0.14 & -0.8 & 0.68 \\ 0.8 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-1$ and $0.4$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ -0.6 & -0.2 & 3.16 \\ -0.5 & 0.5 & 1.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.8 & 1 & 0.4 \\ 0.92 & -0.1 & 1.96 \\ 0.4 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-1$ and $0.4$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ -0.6 & -0.2 & 1.54 \\ -0.5 & 0.5 & 1.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.2 & 4 & 1.6 \\ -0.28 & -1.6 & 1.36 \\ 1.6 & 2 & 0.8 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-0.5$ and $0.2$.

$\left(\begin{array}{rrr} 2.5 & 0 & -2 \\ -1.5 & -1.4 & -0.56 \\ -1.25 & 2 & 3.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.2 & 4 & 2.8 \\ -6.4 & -7 & -4.4 \\ 4.75 & 5 & 3.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-0.5$ and $0.4$.

$\left(\begin{array}{rrr} 5 & 6 & 2 \\ -4.5 & -6.1 & -4.3 \\ -2.5 & -2 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6.6 & 5.6 & 2.72 \\ -5.75 & -5 & -2.5 \\ 1.25 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-2.5$ and $0.4$.

$\left(\begin{array}{rrr} 2 & 0 & -0.4 \\ -1.8 & -0.7 & 0.8 \\ -1 & 1 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.75 & 0.2 & 0.14 \\ -0.5 & -0.8 & 0.44 \\ 1.25 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.5$, $-1$ and $0.8$.

$\left(\begin{array}{rrr} 2 & 0 & -2.8 \\ -1.8 & -0.7 & 2.12 \\ -1 & 1 & 3.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.25 & 1.4 & 0.98 \\ -2 & -2 & -0.4 \\ 1.25 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $5$, $-0.8$ and $0.5$.

$3\times3$ non-symmetric matrices with complex eigenvalues

It seems there are no "nice" invertible $3 \times 3$ matrices that have only one digit beyond the decimal point, so these are some of those that have six or fewer entries with two digits beyond the decimal point. Those with the fewest appear first.

$\left(\begin{array}{rrr} 2 & -2 & -2.8 \\ 1.6 & -2 & 0 \\ -1 & 2 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.4 & 1 & 1.4 \\ 0.32 & 0.3 & 1.12 \\ -0.3 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$ and $1.4\pm0.2j$.

$\left(\begin{array}{rrr} 1 & -1 & -1.4 \\ 0.8 & -1 & 0 \\ -0.5 & 1 & 0.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 2 & 2.8 \\ 0.64 & 0.6 & 2.24 \\ -0.6 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-1$ and $0.7\pm0.1j$.

$\left(\begin{array}{rrr} 5 & 6 & 2 \\ -4.5 & -6.1 & -2.16 \\ -2.5 & -2 & -0.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3.1 & 2.8 & 0.76 \\ -4.5 & -4 & -1.8 \\ 6.25 & 5 & 3.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$ and $0.6\pm0.2j$.

$\left(\begin{array}{rrr} 2.5 & -7.5 & -2 \\ 2 & -5.4 & -1.28 \\ -1.25 & 4.75 & 1.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 1 & 2.4 \\ 1.6 & -1 & 1.6 \\ -5.5 & 5 & -3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-0.6\pm0.8j$ and $-0.5$.

$\left(\begin{array}{rrr} 1 & -1 & -0.8 \\ 0.8 & -1.8 & 2.56 \\ -0.5 & 3 & -2.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 1 & 0.8 \\ -0.16 & 0.6 & 0.64 \\ -0.3 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$ and $0.8\pm0.6j$.

$\left(\begin{array}{rrr} 0.5 & 0 & -1 \\ 0.4 & -2 & 1.4 \\ -0.25 & 5 & -2.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 2 & 0.8 \\ -0.26 & 0.6 & 0.44 \\ -0.6 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$ and $0.5\pm0.5j$.

$\left(\begin{array}{rrr} 0.2 & 0 & -0.4 \\ 0.16 & -0.8 & 0.56 \\ -0.1 & 2 & -1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2 & 5 & 2 \\ -0.65 & 1.5 & 1.1 \\ -1.5 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$ and $0.2\pm0.2j$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ 0.8 & -3.5 & -1.3 \\ -0.5 & 5 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.1 & 2 & 1.4 \\ 0.19 & -0.2 & 0.06 \\ -0.45 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$ and $1\pm j$.

$\left(\begin{array}{rrr} 1 & -3 & 0.4 \\ 0.5 & -1.2 & 1.66 \\ -0.5 & 2 & -1.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2 & 2.5 & 4.5 \\ 0.28 & 0.9 & 1.46 \\ -0.4 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$ and $0.6\pm0.2j$.

$\left(\begin{array}{rrr} 2 & -2 & -1.6 \\ 1 & -2 & 2.4 \\ -1 & 3.5 & -2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.4 & 1 & 0.8 \\ 0.02 & 0.6 & 0.64 \\ -0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$ and $1.4\pm0.2j$.

$\left(\begin{array}{rrr} 1 & 4 & 1.6 \\ -0.6 & -3 & -1.24 \\ -0.5 & 0 & 0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6 & 8 & 0.4 \\ -2.75 & -4 & -0.7 \\ 3.75 & 5 & 1.5 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$ and $0.4\pm0.2j$.

$\left(\begin{array}{rrr} 2 & 8 & 6 \\ -0.2 & -2.8 & -3 \\ -1 & 1 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.05 & 6.5 & 1.8 \\ -0.95 & -3.5 & -1.2 \\ 0.75 & 2.5 & 1 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2\pm j$ and $-0.8$.

$\left(\begin{array}{rrr} 1 & 4 & -0.2 \\ 0.3 & -2.8 & -0.36 \\ -0.5 & 8 & 2.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.6 & 2 & 0.4 \\ 0.09 & -0.4 & -0.06 \\ -0.2 & 2 & 0.8 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$ and $1.4\pm0.2j$.

$\left(\begin{array}{rrr} 1 & -1 & -0.8 \\ 0.5 & -4 & 4.8 \\ -0.5 & 5.5 & -5.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 2 & 1.6 \\ -0.08 & 1.2 & 1.04 \\ -0.15 & 1 & 0.7 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-10$ and $0.7\pm0.1j$.

$\left(\begin{array}{rrr} 0.2 & 0 & -1 \\ 0.16 & -0.4 & 2.8 \\ 0.08 & 1 & -4.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.6 & 2.5 & 1 \\ -2.32 & 2 & 1.8 \\ -0.48 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-5$ and $0.2\pm0.2j$.

$\left(\begin{array}{rrr} 0.1 & 0 & -0.5 \\ 0.08 & -0.2 & 1.4 \\ 0.04 & 0.5 & -2.2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 5.2 & 5 & 2 \\ -4.64 & 4 & 3.6 \\ -0.96 & 1 & 0.4 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2.5$ and $0.1\pm0.1j$.

$\left(\begin{array}{rrr} 2 & -2 & -2.8 \\ 1.6 & -1.3 & -0.42 \\ -1 & 1.5 & 1.1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.4 & 1 & 1.4 \\ 0.67 & 0.3 & 1.82 \\ -0.55 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1.4\pm0.2j$ and $-1$.

$\left(\begin{array}{rrr} 1 & -3 & 3 \\ 0.8 & -4.4 & 6.8 \\ -0.5 & 6.5 & -7.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.12 & 0.3 & 0.72 \\ -0.26 & 0.6 & 0.44 \\ -0.3 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-12.5$ and $0.8\pm0.4j$.

$\left(\begin{array}{rrr} 2 & -6 & -2.8 \\ 1 & -1.5 & 0.9 \\ -1 & 5.5 & 1.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.78 & 0.4 & 0.96 \\ 0.28 & -0.1 & 0.46 \\ -0.4 & 0.5 & -0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $2.2\pm0.4j$ and $-2$.

$\left(\begin{array}{rrr} 2 & -2 & 0.8 \\ 1 & 0.5 & 0.5 \\ -1 & 3.5 & -1.9 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.54 & 0.2 & 0.28 \\ -0.28 & 0.6 & 0.04 \\ -0.8 & 1 & -0.6 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$ and $1.3\pm0.9j$.

$\left(\begin{array}{rrr} 1 & 0 & -2 \\ 0.5 & -0.6 & -0.12 \\ -0.5 & 2 & 1.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.3 & 2 & 0.6 \\ 0.32 & -0.2 & 0.44 \\ -0.35 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $1.4\pm0.2j$ and $-1$.

$\left(\begin{array}{rrr} 1 & -1 & -0.8 \\ 0.5 & -0.8 & 0.96 \\ -0.5 & 1.5 & -0.8 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.8 & 2 & 1.6 \\ 0.08 & 1.2 & 1.36 \\ -0.35 & 1 & 0.3 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-2$ and $0.7\pm0.1j$.

$\left(\begin{array}{rrr} 1 & 0 & -5 \\ 0.5 & -1 & 1.3 \\ -0.5 & 2.5 & -2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.25 & 2.5 & 1 \\ -0.07 & 0.9 & 0.76 \\ -0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-4$ and $1\pm0.5j$.

$\left(\begin{array}{rrr} 0.4 & 0 & -2 \\ 0.2 & -1 & 2.2 \\ -0.2 & 2.5 & -2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.75 & 2.5 & 1 \\ 0.02 & 0.6 & 0.64 \\ -0.15 & 0.5 & 0.2 \\ \end{array}\right)$ with the first matrix having the eigenvalues $-4$ and $0.7\pm0.1j$.

$\left(\begin{array}{rrr} 5 & 3 & 2 \\ -4.5 & -6.2 & -6.3 \\ 1 & 5.6 & 7.4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.06 & 1.1 & 0.65 \\ -2.7 & -3.5 & -2.25 \\ 1.9 & 2.5 & 1.75 \\ \end{array}\right)$ with the first matrix having the eigenvalues $3.5\pm0.5j$ and $-0.8$.

$3\times3$ symmetric matrices

None of these have "nice" eigenvalues, but they all have one nice eigenvalue. You can multiply one matrix by $\pm0.1, \pm0.2, \pm0.4, \pm0.5, \pm2, \pm2.5, \pm5$ or $\pm10$ and then multiply the other by the reciprocal, and the arithmetic will still be "nice".

In some cases, multiples of a matrix are given if that multiple results in one matrix being all integers. The first two are extraordinarily nice, as the three eigenvalues are equally spaced.

$\left(\begin{array}{rrr} 2 & -1 & 1 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 1.5 & 0.5 & -1.5 \\ 0.5 & 0.5 & -0.5 \\ -1.5 & -0.5 & 2.5 \\ \end{array}\right)$ with the first matrix having eigenvalues $2$ and $2 \pm \sqrt{3}$.

$\left(\begin{array}{rrr} 1 & -0.5 & 0.5 \\ -0.5 & 1.5 & 0 \\ 0.5 & 0 & 0.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 3 & 1 & -3 \\ 1 & 1 & -1 \\ -3 & -1 & 5 \\ \end{array}\right)$ with the first matrix having eigenvalues $1$ and $1 \pm 0.5\sqrt{3}$.

$\left(\begin{array}{rrr} 1 & -0.2 & 0.6 \\ -0.2 & 0.2 & -0.2 \\ 0.6 & -0.2 & 0.6 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2.5 & 0 & -2.5 \\ 0 & 7.5 & 2.5 \\ -2.5 & 2.5 & 5 \\ \end{array}\right)$ with the first matrix having eigenvalues $0.2$ and $0.8 \pm 0.4\sqrt{3}$.

$\left(\begin{array}{rrr} 0.5 & -0.1 & 0.3 \\ -0.1 & 0.1 & -0.1 \\ 0.3 & -0.1 & 0.3 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 5 & 0 & -5 \\ 0 & 15 & 5 \\ -5 & 5 & 10 \\ \end{array}\right)$ with the first matrix having eigenvalues $0.1$ and $0.4 \pm 0.4\sqrt{3}$.

$\left(\begin{array}{rrr} 2.5 & -0.5 & -2 \\ -0.5 & 0.9 & 0 \\ -2 & 0 & 2 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.5 & 2.5 & 4.5 \\ 2.5 & 2.5 & 2.5 \\ 4.5 & 2.5 & 5 \\ \end{array}\right)$ with the first matrix having eigenvalues $1$ and $2.2 \pm 0.2\sqrt{111}$.

$\left(\begin{array}{rrr} 0.2 & -0.1 & 0.1 \\ -0.1 & 1.3 & 0.2 \\ 0.1 & 0.2 & 0.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 6.1 & 0.7 & -1.5 \\ 0.7 & 0.9 & -0.5 \\ -1.5 & -0.5 & 2.5 \\ \end{array}\right)$ with the first matrix having eigenvalues $0.5$ and $0.75 \pm 0.05\sqrt{145}$.

$\left(\begin{array}{rrr} 4 & -2 & 2 \\ -2 & 3 & -2 \\ 2 & -2 & 4 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.4 & 0.2 & -0.1 \\ 0.2 & 0.6 & 0.2 \\ -0.1 & 0.2 & 0.4 \\ \end{array}\right)$ with the first matrix having eigenvalues $2$ and $4.5 \pm 0.5 \sqrt{41}$.

$\left(\begin{array}{rrr} 5 & -2.5 & 0.5 \\ -2.5 & 2.5 & -0.5 \\ 0.5 & -0.5 & 0.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 0.4 & 0.4 & 0 \\ 0.4 & 0.9 & 0.5 \\ 0 & 0.5 & 2.5 \\ \end{array}\right)$ with the first matrix having eigenvalues $1$ and $3.5 \pm 0.5\sqrt{39}$.

$\left(\begin{array}{rrr} 0.4 & -0.2 & -0.2 \\ -0.2 & 0.5 & -0.1 \\ -0.2 & -0.1 & 0.7 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 4.25 & 2 & 1.5 \\ 2 & 3 & 1 \\ 1.5 & 1 & 2 \\ \end{array}\right)$ with the first matrix having eigenvalues $0.8$ and $0.4 \pm 0.1\sqrt{6}$.

$\left(\begin{array}{rrr} 5 & -2.5 & -4.5 \\ -2.5 & 2.5 & 2.5 \\ -4.5 & 2.5 & 4.5 \\ \end{array}\right)$ and $\left(\begin{array}{rrr} 2 & 0 & 2 \\ 0 & 0.9 & -0.5 \\ 2 & -0.5 & 2.5 \\ \end{array}\right)$ with the first matrix having eigenvalues $1$ and $5.5 \pm 0.5\sqrt{111}$.