Matrix Calculator (2×2)
Calculate the determinant, inverse, transpose, and eigenvalues of a 2×2 matrix.
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How to use this calculator
For a 2×2 matrix [[a,b],[c,d]]: determinant = ad − bc; inverse exists only when det ≠ 0.
- 1
Enter the four values of your 2×2 matrix: a, b (first row) and c, d (second row).
- 2
The calculator shows determinant, trace, transpose, eigenvalues, and inverse.
- 3
If the determinant is 0, the matrix is singular and has no inverse.
Frequently asked questions
Which matrix operations does this cover?
Only one 2×2 matrix at a time: determinant, trace, transpose, eigenvalues and inverse. It does not add, subtract or multiply two matrices, it does not handle 3×3 or larger sizes, and it does not return eigenvectors.
How do I enter the matrix?
The four boxes follow reading order: a and b are the first row, c and d are the second row, so the matrix is [[a, b], [c, d]]. Entering [[4, 7], [2, 6]] gives a determinant of 10 and trace of 10. Decimals and negative numbers are accepted; enter fractions as decimals.
When does it say there is no inverse?
When the determinant ad − bc equals exactly 0. For [[1, 2], [2, 4]] the determinant is 0 and the output says the matrix is singular. The test is exact, so a determinant that is tiny but not zero produces an inverse with very large entries rather than a warning.
What do the complex eigenvalues mean?
When trace² − 4·det is negative, the real eigenvalues do not exist and the calculator prints the pair as a real part ± an imaginary part with i. The 90° rotation matrix [[0, −1], [1, 0]] returns λ = 0 ± 1i, which makes sense because a pure rotation leaves no real direction unchanged.
How are eigenvalues calculated?
It solves λ² − trace·λ + det = 0 with the quadratic formula. That means the two eigenvalues always add up to the trace and multiply to the determinant. For [[2, 1], [1, 2]] they are 3 and 1: sum 4 equals the trace, product 3 equals the determinant.
Why do I see a repeated eigenvalue for the identity matrix?
[[1, 0], [0, 1]] returns λ₁ = 1 and λ₂ = 1. The root is double because the discriminant is 0. The calculator reports the value twice but does not say whether the matrix has one or two independent eigenvectors.
How many decimals are shown?
Results are rounded to six decimal places, with trailing zeros removed. The inverse of [[2, 0], [0, 3]] shows 0.333333 in the lower right entry, which is an approximation of 1/3. The matrix and transpose lines echo your inputs exactly as typed.
Can I check the inverse myself?
Yes. Multiply the original matrix by the inverse and you should get [[1, 0], [0, 1]]. For [[1, 2], [3, 4]] the calculator shows [[−2, 1], [1.5, −0.5]], and a quick hand multiplication of the first row gives 1·(−2) + 2·1.5 = 1 and 1·1 + 2·(−0.5) = 0.
2×2 matrices: determinant, inverse and eigenvalues
What each output tells you
For a matrix [[a, b], [c, d]] the determinant is ad − bc, the trace is a + d, and the transpose swaps b and c. The determinant is the factor by which the matrix scales area; its sign says whether orientation flips. The inverse, when it exists, is 1/det times [[d, −b], [−c, a]]. The eigenvalues are the roots of λ² − trace·λ + det = 0, so their sum is the trace and their product is the determinant.
| Matrix | Determinant | Trace | Eigenvalues | Inverse |
|---|---|---|---|---|
| [[4, 7], [2, 6]] | 10 | 10 | 8.872983, 1.127017 | [[0.6, −0.7], [−0.2, 0.4]] |
| [[1, 2], [3, 4]] | −2 | 5 | 5.372281, −0.372281 | [[−2, 1], [1.5, −0.5]] |
| [[2, 1], [1, 2]] | 3 | 4 | 3, 1 | [[0.666667, −0.333333], [−0.333333, 0.666667]] |
| [[1, 2], [2, 4]] | 0 | 5 | 5, 0 | Does not exist |
Worked example: an invertible matrix
The default matrix [[4, 7], [2, 6]] has determinant 4·6 − 7·2 = 10, a nonzero value, so it is invertible. The calculator returns [[0.6, −0.7], [−0.2, 0.4]]. To solve 4x + 7y = 1 and 2x + 6y = 0, multiply the inverse by the right-hand side column (1, 0) to read off x = 0.6 and y = −0.2. The eigenvalues, 8.872983 and 1.127017, are both positive and add to the trace of 10.
Worked example: a singular matrix
In [[1, 2], [2, 4]] the second row is twice the first, so the two rows describe the same direction. The determinant is 0, the eigenvalues are 5 and 0, and no inverse exists. Whenever one eigenvalue is 0 the determinant is 0, which is why those two facts always appear together in the output. Geometrically, every point lands on a single line.
Rotations and complex eigenvalues
A matrix with a negative discriminant has no real eigenvalues. The quarter-turn matrix [[0, −1], [1, 0]] has determinant 1, trace 0, and eigenvalues 0 ± 1i. Its inverse is [[0, 1], [−1, 0]], which is the rotation back by the same angle. Complex eigenvalues are the sign that no real direction is merely stretched; the matrix turns vectors instead.
Limits worth knowing
Only one 2×2 matrix is processed per run, so there is no matrix product or sum, and no eigenvectors. Values are rounded to six decimals for display, but the arithmetic itself uses ordinary floating-point numbers. For larger systems or when you need the eigenvectors, use dedicated linear-algebra software.
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