Enter a square matrix and click Calculate Inverse. This tool uses Gauss-Jordan elimination and also verifies the result by multiplying \(A \cdot A^{-1}\).
What Is the Inverse of a Matrix?
For a square matrix A, the inverse matrix is written as A-1. It is defined by the identity:
A · A-1 = I
where I is the identity matrix (1s on the diagonal, 0s elsewhere). If a matrix has an inverse, it is called invertible or non-singular.
When Does an Inverse Exist?
A square matrix has an inverse only if its determinant is not zero:
- If det(A) ≠ 0, the inverse exists.
- If det(A) = 0, the matrix is singular and no inverse exists.
This calculator computes the determinant first and then attempts inversion.
How This Calculator Works
1) Build an augmented matrix
The algorithm creates [A | I], combining your input matrix A with an identity matrix I.
2) Apply Gauss-Jordan elimination
It performs row operations until the left block becomes the identity matrix. At that point, the right block becomes A-1.
3) Verify by multiplication
To reduce mistakes, the tool multiplies A · A-1 and displays the check matrix, which should be very close to identity.
How to Use the Tool
- Choose a matrix size between 2×2 and 6×6.
- Enter every matrix element.
- Click Calculate Inverse.
- Read determinant, inverse matrix, and verification result.
Common Use Cases
- Solving systems of linear equations: Ax = b via x = A-1b
- Computer graphics transformations
- Control systems and state-space models
- Statistics, optimization, and machine learning
- Engineering simulation and signal processing
Tips for Reliable Results
Use precise numbers
Decimals are accepted. Very large or tiny values can amplify floating-point rounding effects.
Avoid nearly singular matrices
If the determinant is extremely close to zero, the inverse can become unstable and contain very large entries.
Cross-check critical calculations
For homework, research, or production workflows, always verify with multiple methods or software.
Quick FAQ
Can I invert a non-square matrix?
No. Standard matrix inversion requires a square matrix.
Why do I get “matrix is singular”?
That means your determinant is zero (or effectively zero). The rows/columns are linearly dependent, so no inverse exists.
Is this method exact?
It is numerically accurate for many practical problems, but all floating-point methods include finite precision effects.