The condition number measures how sensitive the solution of Ax = b is to small changes in A or b. A condition number of 10ᵏ means you can lose about k digits of accuracy – crucial in numerical computing and data fitting.
How to use the Condition Number Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The inverse is computed exactly and both norms are evaluated in the chosen norm. In the 2-norm, cond(A) is the ratio of the largest to the smallest singular value.
Tips
- Check the sizes first: matrix products need the inner dimensions to match.
- Enter the matrix row by row; each row of fields is one row of the matrix.
- Integer matrices are shown with exact fractions where the result is rational.
- A determinant of zero means the matrix has no inverse and its columns are dependent.
Common mistakes to avoid
- Multiplying matrices element by element when a matrix product is needed.
- Assuming AB = BA – matrix multiplication is not commutative.
- Mixing up rows and columns when reading a result.
Formula
cond(A) = ‖A‖ · ‖A⁻¹‖ (∞ if A is singular)
Worked example
Inputs — Matrix size: 2 × 2; a₁: 1; b₁: 2; a₂: 3; b₂: 4; Matrix norm: 2-norm.
Results — Condition number cond(A) = 14.93303437; ‖A‖ = 5.464985704; ‖A⁻¹‖ = 2.732492852; A⁻¹ = ⎡ −2 1 ⎤
⎣ 3/2 −1/2 ⎦.
Key terms
- Ill-conditioned
- Very sensitive to small changes in the data.
Frequently asked questions
What is a good condition number?
Close to 1. Values above about 10¹² are ill-conditioned in double precision.
What does an infinite condition number mean?
The matrix is singular.
Which norm should I use?
The 2-norm is most common; others give similar orders of magnitude.
Is the condition number the same as the determinant?
No – a tiny determinant does not by itself mean ill-conditioning.
What formula does the Condition Number Calculator use?
cond(A) = ‖A‖ · ‖A⁻¹‖ (∞ if A is singular).