What an Inverse Matrix Is
The inverse of a square matrix A is a matrix A⁻¹ that reverses the effect of A. When A⁻¹ exists, multiplying A by A⁻¹ gives the identity matrix: A·A⁻¹ = I. This is the matrix equivalent of division: the identity matrix plays the role of “1” in multiplication.
When a Matrix Is Invertible
A matrix is invertible only if it is square and nonsingular. The most common test is the determinant: if det(A) ≠ 0, an inverse exists. If det(A)=0, the matrix is singular and cannot be inverted. Rank provides the same conclusion for square matrices: full rank means invertible.
How This Matrix Inverse Calculator Works
This tool uses Gauss–Jordan elimination, a standard method for computing inverses numerically. The algorithm constructs an augmented matrix [A|I], then performs row operations until the left side becomes the identity matrix. If that transformation succeeds, the right side becomes A⁻¹.
To improve stability, the calculator uses partial pivoting, which selects a strong pivot in each column. This reduces the risk of dividing by tiny numbers and helps control rounding errors in typical cases.
Why Inverses Can Be Numerically Sensitive
Even when a matrix is technically invertible, it can be ill-conditioned (nearly singular). In those cases, small rounding errors can be amplified into large changes in the computed inverse. This is why the calculator reports det(A) and also includes a verification mode that measures how close A·A⁻¹ is to I.
Using A⁻¹ for Solving Systems
If A is invertible, the system Ax=b has a unique solution x=A⁻¹b. While this is a correct formula, many numerical workflows solve Ax=b directly using elimination without explicitly forming A⁻¹. Inverses are still useful for analysis, transformations, and closed-form derivations, especially in small systems.
How to Use This Calculator
- Choose matrix size n×n (up to 6×6).
- Build the matrix and enter values (or use Quick Fill to start quickly).
- Click Calculate A⁻¹ to generate the inverse and determinant.
- Open Verify to check A·A⁻¹ against the identity matrix.
- Export input or inverse to CSV as needed.
Limitations
This calculator uses floating-point arithmetic in the browser, so exact fraction output is not provided. For symbolic inverses or exact rational results, use a computer algebra system. To keep the tool fast and stable for typical usage, inverse size is limited to 6×6.
FAQ
Matrix Inverse Calculator – Frequently Asked Questions
Answers about invertibility, determinants, Gauss–Jordan elimination, stability, and verifying A⁻¹.
The inverse of a square matrix A is another matrix A⁻¹ such that A·A⁻¹ = I, where I is the identity matrix. It “undoes” the linear transformation defined by A.
A matrix has an inverse only if it is square and nonsingular, meaning det(A) ≠ 0. If det(A)=0, the matrix is singular and not invertible.
It uses Gauss–Jordan elimination on the augmented matrix [A | I]. If the left side reduces to I, the right side becomes A⁻¹. Partial pivoting improves stability.
If a matrix is ill-conditioned, tiny input changes can cause huge changes in the inverse. Floating-point rounding can make the computed inverse unreliable when pivots are extremely small.
No. To keep results stable and fast in the browser, this tool supports inverses up to 6×6. Larger inverses are better handled in specialized software.
If the matrix is singular, the calculator will report that no inverse exists. In many cases, RREF/rank can explain why (dependent rows/columns).
You can multiply A by the computed A⁻¹ and check whether the result is close to the identity matrix (small rounding differences are normal). This tool can also show a quick verification summary.
Yes. You can export the input matrix or the computed inverse to CSV.
No. It uses floating-point arithmetic in the browser. For exact fractions or symbolic inverses, use a computer algebra system.