Matrix Calculator (2×2)
Determinant, inverse, trace and transpose · multiply two 2×2 matrices · eigenvalues, eigenvectors and the characteristic polynomial · solve Ax = b with Cramer's rule — every step and, where exact, the rational form too.
Written by Suhaib Hassan. Every worked example below is re-derived by hand from the formulas. Our methodology.
Determinant and inverse use the standard 2×2 adjugate formula; eigenvalues solve the characteristic quadratic λ² − tr(A)λ + det(A) = 0; the system solver uses Cramer's rule.
Linear algebra homework and quick graphics/engineering checks need exact fractions, not just decimals — this shows both.
—
Geometric effect of A on the unit square
How A maps the basis vectors e₁ = (1, 0) and e₂ = (0, 1).
Step-by-step working
—
Matrix classes at a glance
How a 2×2 matrix can behave.
| Class | Condition | det(A) | Geometric action |
|---|---|---|---|
| Non-singular | rank(A) = 2 | ≠ 0 | Bijective map; area scales by |det(A)| |
| Singular | rank(A) < 2 | = 0 | Collapses the plane onto a line or a point |
| Orthogonal | AᵀA = I | ±1 | Rigid motion — a rotation or reflection |
| Symmetric | A = Aᵀ | — | Real eigenvalues, perpendicular eigenvectors |
Worked examples
Determinant & inverse
For A = [[3, 2], [1, 4]], find det(A) and A⁻¹.
A⁻¹ = (1/10)[[4, −2], [−1, 3]]
= [[0.4, −0.2], [−0.1, 0.3]]
Eigenvalues
Find the eigenvalues of the same matrix A.
λ² − 7λ + 10 = 0 → (λ − 5)(λ − 2) = 0
λ₁ = 5, λ₂ = 2
Cramer's rule
Solve Ax = b for b = (8, 7) with the same A.
x₂ = det([[3,8],[1,7]]) / det(A) = 13/10 = 1.3
x = (1.8, 1.3)
Key 2×2 formulas
- Determinant: det(A) = a·d − b·c for A = [[a, b], [c, d]].
- Inverse: A⁻¹ = (1/det A)·[[d, −b], [−c, a]], defined only when det(A) ≠ 0.
- Trace: tr(A) = a + d, the sum of the diagonal entries.
- Characteristic polynomial: p(λ) = λ² − tr(A)λ + det(A); its roots are the eigenvalues.
- Cramer's rule: for Ax = b, xᵢ = det(Aᵢ) / det(A), where Aᵢ replaces column i of A with b.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London 148, 17–37.
- Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. Chapters 4 & 6.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics, clause 11 (matrices).
Frequently asked questions
What does det(A) = 0 mean?
The rows (or columns) of A are linearly dependent — one is a scalar multiple of the other. The matrix has no inverse, and it maps the plane onto a line or a single point rather than a full 2D region.
How do I check that an inverse is correct?
Multiply A by the computed A⁻¹: the result should be the identity matrix I = [[1, 0], [0, 1]], up to rounding. This calculator performs that check for you.
Can a 2×2 matrix have complex eigenvalues?
Yes — whenever the discriminant tr(A)² − 4·det(A) is negative, such as for a rotation matrix (with angle not a multiple of 180°). The eigenvalues then form a complex conjugate pair.
Why does Cramer's rule fail when det(A) = 0?
Because it divides by det(A). A singular system either has no solution or infinitely many, and Cramer's rule cannot distinguish those cases — use row reduction instead.
What do the eigenvalues tell you geometrically?
Along each eigenvector direction, A acts as pure scaling by the corresponding eigenvalue — no rotation. Real eigenvalues mean the map stretches or flips along fixed directions; complex ones mean the map includes rotation.
Is A⁻¹A always equal to AA⁻¹?
Yes, for a square invertible matrix, A⁻¹A = AA⁻¹ = I. This is a special property of square matrices; it does not hold for non-square matrices, which only have one-sided inverses at best.
Related calculators
Continue with algebra.