Moore-Penrose Pseudoinverse Calculator (2×2)
The Moore-Penrose pseudoinverse A† generalises the concept of a matrix inverse to matrices that are singular or even non-square. Enter any 2×2 matrix and the calculator returns A† along with the matrix rank and a numerical check that A · A† · A = A.
Non-zero → A is invertible; A† = A⁻¹
- 1
a × d
3 × 4 = 12 - 2
b × c
1 × 2 = 2 - 3
Determinant det = ad − bc
12 − 2 = 10A non-zero determinant means the matrix is invertible and A† = A⁻¹.
How does this calculator work?
The Moore-Penrose pseudoinverse A† of a 2×2 matrix equals the ordinary inverse when det ≠ 0, or Aᵀ/‖A‖² when rank = 1. It solves Ax = b in the least-squares sense when no exact solution exists. Enter a, b, c, d and the calculator returns A†, rank, Frobenius norm and a numerical verification.
Formula
How this is calculated
For an invertible 2×2 matrix (det ≠ 0) the pseudoinverse is exactly the ordinary inverse, computed from the adjugate formula: A⁻¹ = (1/det) × [[d, −b], [−c, a]]. No additional processing is needed — the Moore-Penrose conditions are automatically satisfied.
For a singular 2×2 matrix (det = 0 but at least one non-zero entry, i.e. rank 1), the pseudoinverse simplifies to Aᵀ / ‖A‖²_F, where ‖A‖_F is the Frobenius norm √(a²+b²+c²+d²). This comes from the SVD: a rank-1 matrix A = σ u vᵀ has pseudoinverse A† = (1/σ) v uᵀ, and dividing the transpose by the squared Frobenius norm is algebraically equivalent for rank-1 matrices.
The verification row "‖AA†A − A‖" measures how well A† satisfies the first Moore-Penrose condition, A A† A = A. A value close to zero (≤ 10⁻¹⁰) confirms the result is numerically correct. Practical limitations: only 2×2 matrices are supported; extend to larger matrices via SVD libraries in languages like Python (numpy.linalg.pinv) or MATLAB (pinv).
Frequently asked questions
A singular matrix has no ordinary inverse, so you cannot solve Ax = b exactly. A† gives the least-squares solution x = A†b that minimises ‖Ax − b‖², which is the best you can do when the system is under- or over-determined.
Yes. When det(A) ≠ 0 the Moore-Penrose pseudoinverse equals the ordinary inverse A⁻¹. The pseudoinverse generalises the inverse to all matrices, so it is always safe to use A† even when you do not know in advance whether A is invertible.
Four conditions: (1) A A† A = A, (2) A† A A† = A†, (3) (A A†)ᵀ = A A†, (4) (A† A)ᵀ = A† A. The verification row checks condition 1 numerically.
Also known as
TG we-Calculate Editorial Team. (2026). Moore-Penrose Pseudoinverse Calculator (2×2) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pseudoinverse-calculator
TG we-Calculate Editorial Team. "Moore-Penrose Pseudoinverse Calculator (2×2)." TG we-Calculate. 2026. https://we-calculate.com/calculator/pseudoinverse-calculator.
TG we-Calculate Editorial Team, "Moore-Penrose Pseudoinverse Calculator (2×2)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pseudoinverse-calculator
@misc{wecalculate_pseudoinverse_calculator, title = {Moore-Penrose Pseudoinverse Calculator (2×2)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pseudoinverse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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