Calculator Form
Enter a square matrix with one row per line. Separate entries with commas. Supported values include 3, -2.5, i, -i, 4+3i, and 2-0.75i.
Example Data Table
This sample demonstrates a valid 3 × 3 complex matrix ready for inversion.
| Row | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 2 + i | 1 - 2i | 0 |
| R2 | 0 | 3 + i | 2 |
| R3 | 1 + i | 0 | 4 - 3i |
Formula Used
For any invertible square matrix A, the inverse satisfies A × A-1 = I.
In theory, A-1 = adj(A) / det(A) when det(A) ≠ 0. This calculator computes the inverse numerically using complex Gauss–Jordan elimination on the augmented matrix [A | I].
Complex division uses:
(a + bi) / (c + di) = ((ac + bd) / (c² + d²)) + ((bc - ad) / (c² + d²))i
Pivoting selects the largest available complex magnitude in each column. That improves stability and helps identify singular or nearly singular matrices.
How to Use This Calculator
- Enter one matrix row per line.
- Separate entries with commas.
- Use supported complex forms such as
i,-i,3+2i, or4-0.5i. - Choose the display precision and numerical tolerance.
- Select the Plotly view metric you want to inspect.
- Click Compute Inverse.
- Review the determinant, inverse, and verification matrix above the form.
- Use the export buttons to save the inverse as CSV or PDF.
FAQs
1) What makes a complex matrix invertible?
A complex square matrix is invertible when its determinant is nonzero. In elimination terms, every column must produce a valid pivot, allowing the matrix to reduce to the identity while generating the inverse on the augmented side.
2) How should I enter complex numbers?
Use plain forms such as 5, -3.2, i, -i, 4+3i, or 7-0.25i. Enter one row per line and separate entries with commas. Avoid extra symbols like parentheses or multiplication signs.
3) Can this calculator invert non-square matrices?
No. Only square matrices can have a two-sided inverse. If your matrix has different row and column counts, the calculator stops and reports that inversion is not defined for that shape.
4) Why is the determinant displayed?
The determinant quickly shows whether an inverse may exist. A zero determinant means the matrix is singular. Very small determinant magnitudes can also warn that the system is numerically sensitive.
5) What does the verification matrix mean?
It multiplies the original matrix by the computed inverse. The ideal result is the identity matrix. Tiny nonzero off-diagonal values may appear because floating-point arithmetic and rounding are approximate.
6) Why do pivot values matter?
Pivot values show the elimination path used during Gauss–Jordan reduction. Small pivots can indicate poor conditioning, while healthy pivots usually support more stable inversion and more trustworthy numeric output.
7) What does the Plotly graph show?
The graph shows either magnitude, real part, or imaginary part of each inverse entry. It helps you spot large responses, structural patterns, symmetry, and entries that dominate the inverse matrix.
8) Why can results change with precision or tolerance?
Display precision only changes formatting, but tolerance affects pivot decisions. If a matrix is nearly singular, small tolerance changes can alter the elimination path and slightly modify the reported inverse.