How 2x2 matrix calculator works
The 2×2 matrix [[a, b], [c, d]] is the smallest matrix that still does everything matrices do — and everything about it fits on one line. This tool computes the full profile of any 2×2 matrix: the determinant (ad − bc), the trace (a + d), the inverse when one exists, the transpose, the rank, a classification (zero, identity, scalar, diagonal, symmetric, or general), and the eigenvalues — including the complex pair that appears when the discriminant trace² − 4·det goes negative.
The determinant is the headline number. A non-zero determinant means the matrix is invertible: it stretches and rotates space without ever flattening it, and the inverse is [[d, −b], [−c, a]] / det. A zero determinant means the matrix squashes the plane onto a single line — no inverse exists, information is lost, and one of the eigenvalues is necessarily zero. The rank tells you which happened: rank 2 preserves the plane, rank 1 collapses it to a line, rank 0 is the zero matrix.
The eigenvalues come from the quadratic λ² − (trace)λ + det = 0, which is the quadratic equation in disguise — the discriminant decides whether the eigenvectors point along real directions or rotate as complex pairs. A rotation like [[0, −1], [1, 0]] has determinant 1 and trace 0, so its eigenvalues are exactly ±i: nothing in the plane survives a rotation in place, which is why rotations have no real eigenvectors.