The following are essentially all 2 × 2 matrices that have both integer entries and integer eigenvalues that are also symmetric and invertible. No diagonal 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 an invertible matrix that has three repeated eigenvalues, you can search for, for example, "1, 1, 1".

• 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 include diagonal matrices.

Invertible 2 × 2 symmetric matrices with no entries greater than one in absolute value

These are all such matrices up to multiplication by -1, in which case, the eigenvalues are also negated.

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

Invertible 2 × 2 symmetric matrices with no entries greater than two in absolute value

These are all such matrices up to multiplication by -1, in which case, the eigenvalues are also negated; and swapping rows and columns so that the diagonal entries are sorted.

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

Invertible 2 × 2 symmetric matrices with no entries greater than three in absolute value

These are all such matrices up to multiplication by -1, in which case, the eigenvalues are also negated; and swapping rows and columns so that the diagonal entries are sorted. This does not include integer multiples of matrices listed above.

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

Invertible 2 × 2 symmetric matrices with no entries greater than four in absolute value

These are all such matrices up to multiplication by -1, in which case, the eigenvalues are also negated; and swapping rows and columns so that the diagonal entries are sorted. This does not include integer multiples of matrices listed above.

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

Invertible 2 × 2 symmetric matrices with no entries greater than five in absolute value

These are all such matrices up to multiplication by -1, in which case, the eigenvalues are also negated; and swapping rows and columns so that the diagonal entries are sorted. This does not include integer multiples of matrices listed above.

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