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Symmetric 3 × 3 singular matrices

The following are all randomly generated 3 × 3 matrices that have both integer entries and integer eigenvalues. These matrices are symmetric and singular (not invertible). Unless otherwise noted, no two matrices within any one group have the same eigenvalues. They are grouped based on the maximum integer in absolute value in the matrix. The eigenvalues are sorted, so if you want a singular matrix that has three zero eigenvalues, you can search for "0, 0, 0".

No entries greater than one in absolute value
No entries greater than two in absolute value
No entries greater than three in absolute value
No entries greater than four in absolute value
No entries greater than five in absolute value

Also availble are 3 × 3 matrices that are symmetric or invertible or both. While the matrices are in the Matlab format, some of these have been tested in Maple to ensure that they are not the result of numeric error.

Please note, these do not include diagonal matrices.

Singular 3 × 3 symmetric matrices with no entries greater than one in absolute value

Eigenvalues	Matrix
-1, 0, 1	`[0 0 0; 0 0 1; 0 1 0]`
0, 0, 2	`[0 0 0; 0 1 -1; 0 -1 1]`
0, 0, 2	`[0 0 0; 0 1 1; 0 1 1]`
-2, 0, 1	`[0 0 1; 0 0 -1; 1 -1 -1]`
-1, 0, 2	`[0 0 1; 0 0 -1; 1 -1 1]`
-1, 0, 1	`[0 0 1; 0 0 0; 1 0 0]`
-2, 0, 1	`[0 0 1; 0 0 1; 1 1 -1]`
-1, 0, 2	`[0 0 1; 0 0 1; 1 1 1]`
-2, 0, 1	`[0 1 0; 1 -1 -1; 0 -1 0]`
-2, 0, 1	`[0 1 0; 1 -1 1; 0 1 0]`
-1, 0, 1	`[0 1 0; 1 0 0; 0 0 0]`
-1, 0, 2	`[0 1 0; 1 1 -1; 0 -1 0]`
-1, 0, 2	`[0 1 0; 1 1 1; 0 1 0]`
-1, 0, 2	`[1 -1 -1; -1 0 0; -1 0 0]`
0, 0, 3	`[1 -1 -1; -1 1 1; -1 1 1]`
-1, 0, 2	`[1 -1 0; -1 1 0; 0 0 -1]`
0, 0, 2	`[1 -1 0; -1 1 0; 0 0 0]`
0, 1, 2	`[1 -1 0; -1 1 0; 0 0 1]`
-1, 0, 2	`[1 -1 1; -1 0 0; 1 0 0]`
0, 0, 3	`[1 -1 1; -1 1 -1; 1 -1 1]`
-1, 0, 2	`[1 0 -1; 0 -1 0; -1 0 1]`
0, 0, 2	`[1 0 -1; 0 0 0; -1 0 1]`
0, 1, 2	`[1 0 -1; 0 1 0; -1 0 1]`
-2, 0, 1	`[1 0 0; 0 -1 -1; 0 -1 -1]`
-2, 0, 1	`[1 0 0; 0 -1 1; 0 1 -1]`
0, 1, 2	`[1 0 0; 0 1 -1; 0 -1 1]`
0, 1, 2	`[1 0 0; 0 1 1; 0 1 1]`
-1, 0, 2	`[1 0 1; 0 -1 0; 1 0 1]`
0, 0, 2	`[1 0 1; 0 0 0; 1 0 1]`
0, 1, 2	`[1 0 1; 0 1 0; 1 0 1]`
-1, 0, 2	`[1 1 -1; 1 0 0; -1 0 0]`
0, 0, 3	`[1 1 -1; 1 1 -1; -1 -1 1]`
-1, 0, 2	`[1 1 0; 1 1 0; 0 0 -1]`
0, 0, 2	`[1 1 0; 1 1 0; 0 0 0]`
0, 1, 2	`[1 1 0; 1 1 0; 0 0 1]`
-1, 0, 2	`[1 1 1; 1 0 0; 1 0 0]`
0, 0, 3	`[1 1 1; 1 1 1; 1 1 1]`

Singular 3 × 3 symmetric matrices with no entries greater than two in absolute value

Eigenvalues	Matrix
-3, 0, 4	`[ 2 2 2; 2 -2 0; 2 0 1]`
-3, -1, 0	`[-2 -1 1; -1 -1 0; 1 0 -1]`
-4, -2, 0	`[-2 -2 0; -2 -2 0; 0 0 -2]`
0, 1, 3	`[ 2 0 1; 0 0 0; 1 0 2]`
-1, 0, 3	`[ 1 -2 0; -2 1 0; 0 0 0]`
-4, 0, 3	`[-2 -2 2; -2 2 0; 2 0 -1]`
-2, 0, 3	`[ 1 -1 -1; -1 0 2; -1 2 0]`
-4, 0, 2	`[ 0 -2 0; -2 -2 2; 0 2 0]`
-3, 0, 1	`[ 0 -1 1; -1 -2 1; 1 1 0]`
-3, -3, 0	`[-2 1 1; 1 -2 1; 1 1 -2]`
-3, 0, 3	`[ 0 2 -2; 2 -1 0; -2 0 1]`
-2, 0, 2	`[-2 0 0; 0 1 -1; 0 -1 1]`
-2, 0, 4	`[ 0 2 2; 2 1 1; 2 1 1]`
0, 2, 2	`[ 1 1 0; 1 1 0; 0 0 2]`
-3, 0, 2	`[ 0 1 2; 1 -1 -1; 2 -1 0]`
0, 2, 3	`[ 2 0 1; 0 2 -1; 1 -1 1]`
0, 3, 3	`[ 2 -1 1; -1 2 1; 1 1 2]`
-1, 0, 4	`[ 2 2 0; 2 2 0; 0 0 -1]`
-3, -2, 0	`[-2 0 1; 0 -2 -1; 1 -1 -1]`
-4, 0, 0	`[-2 0 2; 0 0 0; 2 0 -2]`
0, 1, 4	`[ 2 -2 0; -2 2 0; 0 0 1]`
0, 0, 6	`[ 2 2 -2; 2 2 -2; -2 -2 2]`
0, 0, 4	`[ 2 0 -2; 0 0 0; -2 0 2]`
-2, -2, 0	`[-1 0 1; 0 -2 0; 1 0 -1]`
-4, 0, 1	`[-2 0 2; 0 1 0; 2 0 -2]`
-4, -1, 0	`[-1 0 0; 0 -2 -2; 0 -2 -2]`
0, 2, 4	`[ 2 2 0; 2 2 0; 0 0 2]`
-6, 0, 0	`[-2 -2 -2; -2 -2 -2; -2 -2 -2]`

Singular 3 × 3 symmetric matrices with no entries greater than three in absolute value

Eigenvalues	Matrix
-2, 0, 7	`[ 2 3 -2; 3 1 -3; -2 -3 2]`
0, 3, 5	`[ 3 1 -2; 1 2 1; -2 1 3]`
-6, 0, 3	`[ 0 0 -3; 0 0 3; -3 3 -3]`
-7, 0, 2	`[-3 3 -2; 3 -3 2; -2 2 1]`
-5, 0, 3	`[-2 3 1; 3 -2 1; 1 1 2]`
-5, -2, 0	`[-2 -1 -2; -1 -3 -1; -2 -1 -2]`
-5, 0, 2	`[ 0 0 -1; 0 0 -3; -1 -3 -3]`
-6, 0, 2	`[-1 2 -3; 2 -2 2; -3 2 -1]`
-8, 0, 3	`[-3 3 3; 3 1 -3; 3 -3 -3]`
-6, -2, 0	`[-2 -2 -2; -2 -3 -1; -2 -1 -3]`
-2, 0, 5	`[ 0 3 0; 3 3 -1; 0 -1 0]`
0, 3, 4	`[ 2 2 0; 2 2 0; 0 0 3]`
-4, 0, 5	`[ 1 -1 3; -1 1 -3; 3 -3 -1]`
0, 3, 6	`[ 3 0 0; 0 3 3; 0 3 3]`
-5, 0, 4	`[-2 -2 2; -2 -2 2; 2 2 3]`
-5, -1, 0	`[-3 2 0; 2 -3 0; 0 0 0]`
-3, 0, 6	`[ 3 -3 0; -3 3 0; 0 0 -3]`
-1, 0, 6	`[ 3 0 -3; 0 -1 0; -3 0 3]`
0, 0, 9	`[ 3 3 3; 3 3 3; 3 3 3]`
0, 2, 6	`[ 3 2 -1; 2 2 -2; -1 -2 3]`
-5, -3, 0	`[-2 -1 1; -1 -3 -2; 1 -2 -3]`
-6, 0, 1	`[-3 0 3; 0 1 0; 3 0 -3]`
-6, 0, 5	`[-2 -2 -3; -2 -2 -3; -3 -3 3]`
-2, 0, 6	`[ 1 2 3; 2 2 2; 3 2 1]`
-4, -3, 0	`[-3 -1 1; -1 -1 -1; 1 -1 -3]`
-1, 0, 5	`[ 2 0 -3; 0 0 0; -3 0 2]`
-3, 0, 8	`[ 3 -3 3; -3 3 -3; 3 -3 -1]`
-5, 0, 6	`[ 2 3 2; 3 -3 3; 2 3 2]`
0, 2, 5	`[ 3 1 1; 1 2 2; 1 2 2]`
-3, 0, 5	`[ 2 -1 -3; -1 -2 -1; -3 -1 2]`
0, 1, 6	`[ 3 0 -3; 0 1 0; -3 0 3]`
0, 1, 5	`[ 0 0 0; 0 3 2; 0 2 3]`
-6, -1, 0	`[-3 -3 0; -3 -3 0; 0 0 -1]`
-6, -3, 0	`[-3 3 0; 3 -3 0; 0 0 -3]`
-5, 0, 1	`[-2 3 0; 3 -2 0; 0 0 0]`
-9, 0, 0	`[-3 -3 -3; -3 -3 -3; -3 -3 -3]`

Singular 3 × 3 symmetric matrices with no entries greater than four in absolute value

Eigenvalues	Matrix
-4, 0, 8	`[ 0 -4 0; -4 4 -4; 0 -4 0]`
-5, 0, 5	`[ 0 0 4; 0 0 -3; 4 -3 0]`
-8, 0, 4	`[-2 -2 4; -2 -2 4; 4 4 0]`
0, 4, 5	`[ 4 0 0; 0 4 2; 0 2 1]`
-7, 0, 5	`[-4 -4 2; -4 3 2; 2 2 -1]`
-6, -4, 0	`[-3 -2 -1; -2 -4 2; -1 2 -3]`
-4, 0, 6	`[ 0 0 0; 0 -2 4; 0 4 4]`
-8, -1, 0	`[-1 0 0; 0 -4 4; 0 4 -4]`
-5, 0, 7	`[ 4 2 -4; 2 1 -2; -4 -2 -3]`
-6, 0, 4	`[-2 -2 -2; -2 0 -4; -2 -4 0]`
-8, -2, 0	`[-3 -3 -2; -3 -3 -2; -2 -2 -4]`
0, 3, 7	`[ 4 3 -1; 3 4 1; -1 1 2]`
-5, 0, 9	`[-3 2 4; 2 4 4; 4 4 3]`
0, 0, 12	`[ 4 4 4; 4 4 4; 4 4 4]`
-9, 0, 5	`[ 3 -2 4; -2 -4 4; 4 4 -3]`
-10, 0, 1	`[-4 4 -3; 4 -4 3; -3 3 -1]`
-7, 0, 1	`[-3 -4 0; -4 -3 0; 0 0 0]`
-3, 0, 7	`[ 0 -2 -1; -2 0 4; -1 4 4]`
0, 2, 9	`[ 4 -2 3; -2 4 -3; 3 -3 3]`
-1, 0, 7	`[ 0 0 0; 0 3 4; 0 4 3]`
-6, 0, 6	`[-1 1 4; 1 -1 -4; 4 -4 2]`
0, 4, 6	`[ 4 2 0; 2 2 -2; 0 -2 4]`
-5, -4, 0	`[-1 0 2; 0 -4 0; 2 0 -4]`
-1, 0, 8	`[ 4 4 0; 4 4 0; 0 0 -1]`
0, 2, 8	`[ 4 0 4; 0 2 0; 4 0 4]`
-8, 0, 6	`[-2 0 -4; 0 4 4; -4 4 -4]`
-1, 0, 10	`[ 4 4 3; 4 4 3; 3 3 1]`
-4, -4, 0	`[-4 0 0; 0 -2 -2; 0 -2 -2]`
-7, 0, 3	`[ 0 -2 1; -2 0 4; 1 4 -4]`
-8, 0, 0	`[ 0 0 0; 0 -4 4; 0 4 -4]`
-2, 0, 10	`[ 3 -3 -4; -3 3 4; -4 4 2]`
-9, 0, 2	`[-3 -3 3; -3 -2 4; 3 4 -2]`
-7, -3, 0	`[-2 1 1; 1 -4 3; 1 3 -4]`
-4, 0, 4	`[ 2 -2 0; -2 2 0; 0 0 -4]`
-5, 0, 0	`[ 0 0 0; 0 -1 2; 0 2 -4]`
0, 4, 8	`[ 4 0 -4; 0 4 0; -4 0 4]`
0, 1, 8	`[ 1 0 0; 0 4 -4; 0 -4 4]`
-6, -6, 0	`[-4 2 -2; 2 -4 -2; -2 -2 -4]`
0, 0, 5	`[ 0 0 0; 0 4 2; 0 2 1]`
-6, 0, 8	`[ 2 4 0; 4 4 -4; 0 -4 -4]`
0, 6, 6	`[ 4 -2 2; -2 4 2; 2 2 4]`
0, 1, 7	`[ 4 -3 0; -3 4 0; 0 0 0]`
-10, 0, 2	`[-3 -3 -4; -3 -3 -4; -4 -4 -2]`
-9, -2, 0	`[-4 3 2; 3 -3 -3; 2 -3 -4]`
-2, 0, 9	`[ 2 -4 -3; -4 2 3; -3 3 3]`
-7, -1, 0	`[ 0 0 0; 0 -4 -3; 0 -3 -4]`
-8, -3, 0	`[-4 0 -4; 0 -3 0; -4 0 -4]`
0, 3, 8	`[ 3 0 0; 0 4 -4; 0 -4 4]`
-8, -4, 0	`[-4 0 4; 0 -4 0; 4 0 -4]`
-2, 0, 8	`[ 4 4 0; 4 4 0; 0 0 -2]`
-8, 0, 2	`[-4 0 4; 0 2 0; 4 0 -4]`
-8, 0, 1	`[-4 0 -4; 0 1 0; -4 0 -4]`
0, 0, 8	`[ 0 0 0; 0 4 -4; 0 -4 4]`
0, 4, 4	`[ 4 0 0; 0 2 -2; 0 -2 2]`
-12, 0, 0	`[-4 -4 4; -4 -4 4; 4 4 -4]`

Singular 3 × 3 symmetric matrices with no entries greater than five in absolute value

Eigenvalues	Matrix
-4, 0, 11	`[ 1 -5 -5; -5 3 3; -5 3 3]`
0, 2, 11	`[ 4 -3 -4; -3 5 3; -4 3 4]`
-7, 0, 8	`[-1 -1 5; -1 -1 5; 5 5 3]`
-9, 0, 1	`[-5 -4 -2; -4 -3 -2; -2 -2 0]`
-7, 0, 4	`[ 1 -3 1; -3 -5 -3; 1 -3 1]`
-1, 0, 9	`[ 0 -2 2; -2 5 -4; 2 -4 3]`
-4, 0, 9	`[ 1 5 -3; 5 1 -3; -3 -3 3]`
0, 5, 8	`[ 4 0 -4; 0 5 0; -4 0 4]`
-12, 0, 2	`[-3 4 -5; 4 -4 4; -5 4 -3]`
-4, 0, 7	`[-1 -1 3; -1 -1 3; 3 3 5]`
-8, 0, 7	`[ 1 -1 -5; -1 1 5; -5 5 -3]`
-4, 0, 10	`[-4 0 0; 0 5 5; 0 5 5]`
0, 5, 9	`[ 5 0 2; 0 5 4; 2 4 4]`
-6, 0, 12	`[-4 4 -4; 4 5 -5; -4 -5 5]`
0, 2, 12	`[ 5 4 3; 4 4 4; 3 4 5]`
-9, -4, 0	`[-5 -2 4; -2 -4 0; 4 0 -4]`
-9, -1, 0	`[-5 4 0; 4 -5 0; 0 0 0]`
-6, 0, 9	`[ 2 4 -4; 4 5 -2; -4 -2 -4]`
-7, -2, 0	`[-2 0 -1; 0 -2 -3; -1 -3 -5]`
-6, -5, 0	`[-5 0 -2; 0 -5 -1; -2 -1 -1]`
-12, -2, 0	`[-4 -4 -4; -4 -5 -3; -4 -3 -5]`
-1, 0, 11	`[ 4 3 -5; 3 2 -3; -5 -3 4]`
0, 5, 10	`[ 5 0 -3; 0 5 4; -3 4 5]`
-7, -4, 0	`[-5 -1 -1; -1 -3 -3; -1 -3 -3]`
0, 1, 12	`[ 3 3 -3; 3 5 -5; -3 -5 5]`
-9, 0, 4	`[-3 3 3; 3 -1 -5; 3 -5 -1]`
0, 5, 6	`[ 5 -1 0; -1 1 2; 0 2 5]`
-15, 0, 0	`[-5 -5 5; -5 -5 5; 5 5 -5]`
-11, -2, 0	`[-4 -3 -4; -3 -5 -3; -4 -3 -4]`
-11, -1, 0	`[-5 3 4; 3 -2 -3; 4 -3 -5]`
-11, 0, 3	`[-5 4 -4; 4 0 4; -4 4 -3]`
-2, 0, 12	`[ 3 5 4; 5 3 4; 4 4 4]`
-3, 0, 11	`[ 5 4 -4; 4 0 -4; -4 -4 3]`
-10, -5, 0	`[-5 0 4; 0 -5 -3; 4 -3 -5]`
-9, -5, 0	`[-4 4 2; 4 -5 0; 2 0 -5]`
-9, 0, 6	`[-2 2 5; 2 -2 -5; 5 -5 1]`
-12, -1, 0	`[-5 5 -3; 5 -5 3; -3 3 -3]`
-10, -1, 0	`[-5 0 -5; 0 -1 0; -5 0 -5]`
-7, -5, 0	`[-5 0 1; 0 -5 3; 1 3 -2]`
0, 4, 9	`[ 5 2 -4; 2 4 0; -4 0 4]`
-8, -6, 0	`[-5 3 -2; 3 -5 -2; -2 -2 -4]`
-10, 0, 5	`[-5 -5 -5; -5 0 0; -5 0 0]`
0, 0, 15	`[ 5 5 5; 5 5 5; 5 5 5]`
0, 2, 10	`[ 2 0 0; 0 5 -5; 0 -5 5]`
0, 1, 11	`[ 5 3 -4; 3 2 -3; -4 -3 5]`
-9, 0, 3	`[-4 -1 -5; -1 2 1; -5 1 -4]`
-13, 0, 2	`[-3 -5 -5; -5 -4 -4; -5 -4 -4]`
0, 0, 10	`[ 5 0 -5; 0 0 0; -5 0 5]`
0, 3, 9	`[ 5 4 -1; 4 5 1; -1 1 2]`
-5, 0, 10	`[ 0 -5 0; -5 5 5; 0 5 0]`
0, 3, 10	`[ 5 0 -5; 0 3 0; -5 0 5]`
-10, 0, 0	`[-5 5 0; 5 -5 0; 0 0 0]`
0, 5, 7	`[ 5 0 1; 0 5 3; 1 3 2]`
0, 2, 7	`[ 5 -1 -3; -1 2 0; -3 0 2]`
-2, 0, 13	`[ 4 -4 5; -4 4 -5; 5 -5 3]`
-11, 0, 4	`[-3 3 -5; 3 -3 5; -5 5 -1]`
0, 5, 5	`[ 5 0 0; 0 1 2; 0 2 4]`
-3, 0, 9	`[ 4 -5 -1; -5 4 -1; -1 -1 -2]`
-10, -4, 0	`[-5 5 0; 5 -5 0; 0 0 -4]`
-9, -3, 0	`[-2 1 -1; 1 -5 -4; -1 -4 -5]`
0, 1, 10	`[ 1 0 0; 0 5 -5; 0 -5 5]`
-10, -2, 0	`[-2 0 0; 0 -5 5; 0 5 -5]`
-5, -5, 0	`[-5 0 0; 0 -4 -2; 0 -2 -1]`
0, 1, 9	`[ 5 0 -4; 0 0 0; -4 0 5]`
-12, 0, 6	`[-5 -4 -5; -4 4 -4; -5 -4 -5]`
-11, 0, 1	`[-4 -5 -3; -5 -4 -3; -3 -3 -2]`
0, 4, 7	`[ 5 1 -1; 1 3 -3; -1 -3 3]`
-5, 0, 8	`[-5 0 0; 0 4 -4; 0 -4 4]`
0, 6, 8	`[ 4 -2 2; -2 5 3; 2 3 5]`
-8, 0, 5	`[-4 0 -4; 0 5 0; -4 0 -4]`
-8, -5, 0	`[-4 0 4; 0 -5 0; 4 0 -4]`
-10, 0, 3	`[-5 5 0; 5 -5 0; 0 0 3]`
-3, 0, 10	`[ 5 5 0; 5 5 0; 0 0 -3]`
0, 4, 10	`[ 4 0 0; 0 5 5; 0 5 5]`
-10, -3, 0	`[-5 -5 0; -5 -5 0; 0 0 -3]`
-10, 0, 4	`[-5 5 0; 5 -5 0; 0 0 4]`