2×2 Matrix Calculator — Determinant, Inverse, Eigenvalues
Enter a 2×2 matrix and instantly get its determinant, trace, rank, eigenvalues, and inverse — all in one calculator, with an animated visualization of the linear transformation.
Invertible — the matrix has a unique inverse
- 1
a × d
3 × 4 = 12 - 2
b × c
2 × 1 = 2 - 3
det(A) = (a × d) − (b × c)
12 − 2 = 10The determinant measures the signed area scale factor of the linear transformation.
Come funziona questo calcolatore?
For A = [[a,b],[c,d]]: det = ad−bc, trace = a+d, eigenvalues solve λ²−(a+d)λ+(ad−bc)=0, inverse = (1/det)[[d,−b],[−c,a]] if det≠0. Enter your four numbers to get all properties plus a live visualization of the linear transformation.
Formula
How this is calculated
For a 2×2 matrix A = [[a, b], [c, d]], the four core scalar quantities are: the determinant det(A) = ad − bc (the signed area scale factor of the linear map), the trace tr(A) = a + d (sum of diagonal entries), the rank (2 if det ≠ 0, 1 if at least one non-zero entry, 0 for the zero matrix), and the eigenvalues — the scalars λ that satisfy Av = λv for some non-zero vector v.
Eigenvalues solve the characteristic equation det(A − λI) = 0, which simplifies to λ² − tr(A)·λ + det(A) = 0. The quadratic formula gives λ = (tr ± √(tr² − 4·det)) / 2. When the discriminant tr² − 4·det is negative the eigenvalues are complex conjugates; otherwise they are real (and equal when the discriminant is zero).
The inverse exists only when det ≠ 0 and equals A⁻¹ = (1/det) × [[d, −b], [−c, a]]. The animated grid shows how the unit square deforms under the linear transformation — the area of the transformed parallelogram equals |det(A)|.
Domande frequenti
A zero determinant means the matrix is singular: it collapses space into fewer dimensions (a line or point), has no inverse, and the associated linear system either has infinitely many solutions or none. The rank is at most 1 and the transformation squashes area to zero.
Complex eigenvalues come in conjugate pairs of the form α ± βi. They indicate that the transformation has a rotational component. The real part α controls scaling (stretching or shrinking) and the imaginary part β controls rotation. Real matrices with complex eigenvalues have no real eigenvectors.
The standard 2×2 inverse formula A⁻¹ = (1/det) × [[d, −b], [−c, a]] swaps the diagonal entries a and d, negates the off-diagonal entries b and c, then divides everything by the determinant. Make sure you are reading rows in the correct order: row 1 is [d/det, −b/det] and row 2 is [−c/det, a/det].
Conosciuto anche come
TG we-Calculate Editorial Team. (2026). 2×2 Matrix Calculator — Determinant, Inverse, Eigenvalues [Online calculator]. TG we-Calculate. https://we-calculate.com/it/calculator/matrix-calculator
TG we-Calculate Editorial Team. "2×2 Matrix Calculator — Determinant, Inverse, Eigenvalues." TG we-Calculate. 2026. https://we-calculate.com/it/calculator/matrix-calculator.
TG we-Calculate Editorial Team, "2×2 Matrix Calculator — Determinant, Inverse, Eigenvalues," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/it/calculator/matrix-calculator
@misc{wecalculate_matrix_calculator, title = {2×2 Matrix Calculator — Determinant, Inverse, Eigenvalues}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/it/calculator/matrix-calculator}}, year = {2026}, note = {TG we-Calculate} }
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