Inverse Matrix Calculator
With the condition number, so you know what it is worth.
Work out Inverse Matrix. With the condition number, so you know what it is worth. Refuses out-of-range input instead of guessing.
Result
Invertible
Determinant 35 · condition estimate 2.06
Computed by Gauss-Jordan elimination on the augmented matrix, with partial pivoting — rows are swapped so the largest available entry is used as each pivot, which is what keeps rounding error from compounding. The condition estimate is the ratio of the largest pivot to the smallest, a cheap stand-in for the true condition number. Above about 1e10 the inverse has lost most of its significant figures, and printing sixteen digits of it would be printing noise. For solving a system of equations, do not compute the inverse and multiply. Gaussian elimination on the system directly is both faster and numerically better behaved, and the inverse is genuinely needed far less often than it is used.
How the Inverse Matrix Calculator works
The inverse of a 2×2 or 3×3 matrix by Gauss-Jordan elimination with partial pivoting, reported alongside a condition estimate — because an inverse computed from a near-zero determinant is sixteen digits of noise.
Also known as: invert a 3x3 matrix · matrix inverse step by step · why has my matrix no inverse · gauss jordan inverse
Frequently asked questions
When does a matrix have no inverse?
When its determinant is zero. Geometrically the transformation collapses space onto a line or a plane, and information destroyed that way cannot be recovered — there is nothing to invert.
What does the condition number tell me?
How much the inverse amplifies error. Above about 1e10 the result has lost most of its significant figures, so small changes in the input produce large changes in the output and the answer should be treated with caution.
Should I use the inverse to solve equations?
No. Gaussian elimination on the system directly is both faster and numerically better behaved. Computing an inverse in order to multiply by it is one of the most common inefficiencies in applied linear algebra.
What is partial pivoting?
Swapping rows so the largest available entry is used as each pivot. Without it, a small leading coefficient becomes a divisor and amplifies rounding error through every subsequent row — it is not an optimisation but a correctness requirement.
How do I check an inverse is right?
Multiply it by the original matrix. You should get the identity to within rounding error, and if you do not, the matrix is either singular or badly conditioned.
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