The following are essentially all 2 × 2 matrices that have both integer entries and integer eigenvalues that are also non-symmetric and singular (not invertible). No lower-triangular nor upper-triangular matrices are listed, as the eigenvalues of these are obvious. All other cases are listed up to multiplication by -1, so the complementary matrix may be found by multiplying a listed matrix by -1, in which case, the corresponding complementary eigenvalues are the corresponding eigenvalues negated. 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 2 × 2 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 contain upper- or lower-triangular matrices.

Singular 2 × 2 non-symmetric matrices with no entries greater than one in absolute value

Eigenvalues	Matrix
0, 0	[1 -1;  1 -1]
0, 0	[1  1; -1 -1]

Singular 2 × 2 non-symmetric matrices with no entries greater than two in absolute value

These do not include integer multiples of matrices already listed above.

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

Singular 2 × 2 non-symmetric matrices with no entries greater than three in absolute value

These do not include integer multiples of matrices already listed above.

Eigenvalues	Matrix
0, 4	[1 -3; -1  3]
-2, 0	[1 -3;  1 -3]
0, 4	[1 -1; -3  3]
-2, 0	[1 -1;  3 -3]
-2, 0	[1  1; -3 -3]
0, 4	[1  1;  3  3]
-2, 0	[1  3; -1 -3]
0, 4	[1  3;  1  3]
0, 5	[2 -3; -2  3]
-1, 0	[2 -3;  2 -3]
0, 5	[2 -2; -3  3]
-1, 0	[2 -2;  3 -3]
-1, 0	[2  2; -3 -3]
0, 5	[2  2;  3  3]
-1, 0	[2  3; -2 -3]
0, 5	[2  3;  2  3]
0, 5	[3 -3; -2  2]
0, 4	[3 -3; -1  1]
0, 2	[3 -3;  1 -1]
0, 1	[3 -3;  2 -2]
0, 5	[3 -2; -3  2]
0, 1	[3 -2;  3 -2]
0, 4	[3 -1; -3  1]
0, 2	[3 -1;  3 -1]
0, 2	[3  1; -3 -1]
0, 4	[3  1;  3  1]
0, 1	[3  2; -3 -2]
0, 5	[3  2;  3  2]
0, 1	[3  3; -2 -2]
0, 2	[3  3; -1 -1]
0, 4	[3  3;  1  1]
0, 5	[3  3;  2  2]

Singular 2 × 2 non-symmetric matrices with no entries greater than four in absolute value

These do not include integer multiples of matrices already listed above.

Eigenvalues	Matrix
0, 5	[1 -4; -1  4]
-3, 0	[1 -4;  1 -4]
-3, 0	[1 -2;  2 -4]
0, 5	[1 -1; -4  4]
-3, 0	[1 -1;  4 -4]
-3, 0	[1  1; -4 -4]
0, 5	[1  1;  4  4]
-3, 0	[1  2; -2 -4]
-3, 0	[1  4; -1 -4]
0, 5	[1  4;  1  4]
0, 4	[2 -4; -1  2]
0, 0	[2 -4;  1 -2]
0, 4	[2 -1; -4  2]
0, 0	[2 -1;  4 -2]
0, 0	[2  1; -4 -2]
0, 4	[2  1;  4  2]
0, 0	[2  4; -1 -2]
0, 4	[2  4;  1  2]
0, 7	[3 -4; -3  4]
-1, 0	[3 -4;  3 -4]
0, 7	[3 -3; -4  4]
-1, 0	[3 -3;  4 -4]
-1, 0	[3  3; -4 -4]
0, 7	[3  3;  4  4]
-1, 0	[3  4; -3 -4]
0, 7	[3  4;  3  4]
0, 7	[4 -4; -3  3]
0, 5	[4 -4; -1  1]
0, 3	[4 -4;  1 -1]
0, 1	[4 -4;  3 -3]
0, 7	[4 -3; -4  3]
0, 1	[4 -3;  4 -3]
0, 3	[4 -2;  2 -1]
0, 5	[4 -1; -4  1]
0, 3	[4 -1;  4 -1]
0, 3	[4  1; -4 -1]
0, 5	[4  1;  4  1]
0, 3	[4  2; -2 -1]
0, 1	[4  3; -4 -3]
0, 7	[4  3;  4  3]
0, 1	[4  4; -3 -3]
0, 3	[4  4; -1 -1]
0, 5	[4  4;  1  1]
0, 7	[4  4;  3  3]

Singular 2 × 2 non-symmetric matrices with no entries greater than five in absolute value

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