2×2 Matrix Calculator
Determinant, inverse, product and eigenvalues of 2×2 matrices.
Overview
The 2x2 Matrix Calculator gives you the four most-asked operations for small square matrices: determinant, inverse, matrix product and eigenvalues. It is small enough to be useful in a hurry — paste in four numbers, get a clean breakdown without firing up NumPy or MATLAB.
Linear-algebra students checking exam answers, graphics programmers debugging transforms and economists doing simple equilibrium models will all find a use for it. The eigenvalue solver is handy for stability analysis of a 2D dynamical system.
How it works
For a matrix M = [[a, b], [c, d]] the determinant is det(M) = a*d - b*c. When det != 0 the inverse is M^-1 = (1 / det) * [[d, -b], [-c, a]]. The product of two 2x2 matrices follows the standard row-times-column rule.
Eigenvalues solve the characteristic equation λ^2 - tr(M) * λ + det(M) = 0 where tr(M) = a + d. The quadratic formula gives λ = (tr +/- sqrt(tr^2 - 4*det)) / 2. A negative discriminant produces a complex conjugate pair, which the tool reports as α +/- β i.
Examples
det of [[1, 2], [3, 4]] → -2
inverse of [[4, 7], [2, 6]] → [[0.6, -0.7], [-0.2, 0.4]]
[[1, 2], [3, 4]] * [[2, 0], [1, 2]] → [[4, 4], [10, 8]]
eigenvalues of [[2, 1], [1, 2]] → 3, 1
FAQ
What if the matrix is singular?
When the determinant is zero, no inverse exists. The tool flags this rather than dividing by zero.
Can it handle complex eigenvalues?
Yes — a negative discriminant in the characteristic equation produces a complex conjugate pair, reported in α +/- β i form.
Does it support symbolic entries?
No, all entries must be numeric. For symbolic linear algebra reach for a CAS.
Are eigenvectors returned too?
This tool focuses on eigenvalues. For each eigenvalue λ, the eigenvector satisfies (M - λI) v = 0, easy to solve by hand once you have λ.
Is the inverse exact?
It is exact up to double-precision rounding. Integer matrices with integer inverses come back exactly when no division is needed.