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2×2 Moore–Penrose Pseudoinverse Calculator — result sheet
Calculate a transparent Moore–Penrose pseudoinverse for a real 2×2 matrix, including full-rank, rank-one, and zero-matrix cases.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For full rank, A⁺=A⁻¹. For a nonzero rank-one 2×2 matrix, A⁺=Aᵀ/||A||²_F; the zero matrix maps to the zero matrix. The tolerance controls numerical rank classification.
The ordinary inverse exists only for a nonsingular square matrix. The Moore–Penrose pseudoinverse extends the idea to singular matrices, so this compact 2×2 tool exposes the detected rank and the branch used instead of failing on a zero determinant.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Matrix entry a₁₁ — minimum -1000000; maximum 1000000
- Matrix entry a₁₂ — minimum -1000000; maximum 1000000
- Matrix entry a₂₁ — minimum -1000000; maximum 1000000
- Matrix entry a₂₂ — minimum -1000000; maximum 1000000
- Rank tolerance — minimum 1.0E-15; maximum 1
Worked example
| Input | Value |
|---|---|
| Matrix entry a₁₁ | 1 |
| Matrix entry a₁₂ | 2 |
| Matrix entry a₂₁ | 2 |
| Matrix entry a₂₂ | 4 |
| Rank tolerance | 1.0E-10 |
The matrix has rank 1; A⁺ = [[0.04, 0.08], [0.08, 0.16]].
Assumptions and limits
- The matrix has real finite entries and is limited to 2×2 form.
- Rank is classified with the entered tolerance; near-singular results can change when the tolerance changes.
- For a full-rank matrix the pseudoinverse equals the ordinary inverse.
- For a nonzero rank-one 2×2 matrix the transpose divided by the squared Frobenius norm is the exact rank-one pseudoinverse.
- The zero matrix has the zero matrix as its pseudoinverse.
- This page does not compute a larger SVD, covariance regularization, least-squares model, or conditioning report.
- A numerically small determinant is not automatically physically singular; interpret the tolerance and matrix scale together.
- Use a stable numerical linear-algebra library for large matrices or high-stakes scientific computation.
- The displayed matrix entries are decimal approximations when the inputs are decimal values.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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