2×2 Matrix Pseudoinverse
2×2 Matrix Pseudoinverse with stated conventions, validation and a worked example.
Formula
- Use the units shown beside each field.
- Results update automatically whenever you change an input.
- Displayed values use up to four decimal places, or scientific notation for very small or large numbers. Calculations use standard floating-point arithmetic; exact integer and fraction tools identify their own precision rules.
- Real matrices only, at most 6 rows and 6 columns; each entry has magnitude at most 10⁹. Numerical rank uses a relative pivot threshold of 10⁻¹² after scaling by the largest entry. Near-singular matrices may be rejected or treated as lower rank. Displayed entries use 12 significant digits. Rank-one closed form is used only when the scaled determinant is exactly zero in arithmetic; uncertain near-zero cases are rejected.
Example Calculation
- Matrix: 1, 2; 2, 4
For [[1,2],[2,4]], ||A||F²=25 and A⁺=[[0.04,0.08],[0.08,0.16]].
Method references
Frequently asked questions
How do I use this calculator?
Enter matrix. The result updates automatically. Use Reset to restore the example values.
What method does it use?
Full rank: A⁺=A⁻¹. Exact rank one: A⁺=Aᵀ/||A||F². Zero matrix: A⁺=0.. Real matrices only, at most 6 rows and 6 columns; each entry has magnitude at most 10⁹. Numerical rank uses a relative pivot threshold of 10⁻¹² after scaling by the largest entry. Near-singular matrices may be rejected or treated as lower rank. Displayed entries use 12 significant digits. Rank-one closed form is used only when the scaled determinant is exactly zero in arithmetic; uncertain near-zero cases are rejected.
Why might a rounded result differ?
The calculation keeps full numeric precision internally, then rounds the displayed result. Rounding intermediate steps by hand can produce a slightly different answer.
Are my inputs saved online?
No. This calculator processes your inputs in your browser. A recently used list stores only calculator names on this device, not your entered values.
Check the assumptions and units before using this result. Read about calculation methods.