Understand the subject
What is least squares?
Least squares chooses the solution whose output Ax is closest to b when an exact solution is unavailable.
Use it for an overdetermined full-column-rank linear model.
See the structure
What the calculation is doing
[ 1 1 | 3 ]R₂ − 2R₁[ 2 1 | 4 ]→pivots reveal the solution
Worked interpretation
Read the result in context
x̂=(AᵀA)⁻¹Aᵀb
Interpret with care
Important boundary
It reports a mathematical best fit, not proof that a model is appropriate.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Set the matrix dimensions required by the task.
- Fill every visible entry and any operation-specific control.
- Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
x̂ = (AᵀA)⁻¹Aᵀb
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
The residual satisfies Aᵀ(Ax̂−b)=0.
x̂ = (AᵀA)⁻¹Aᵀb
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Exactness
Finite decimals become reduced rational numbers and intermediate structural calculations are not rounded.
Supported scope
This contract requires full column rank; rank-deficient minimum-norm solutions belong to the pseudoinverse calculator.
