Input Matrix A (4×4)
Enter complex numbers like 3+2i, -4i, 5, or rationals like 1/3, 2/5+i/7. Spaces allowed. Use comma-paste from spreadsheets.
Results
Status: –
Max |AI − I|: –
Example Data Table
| Row | Values (use Load Example to fill inputs) |
|---|---|
| 1 | 2+i/2, 0, -1, 1/3 |
| 2 | 3-2i, i, 4, 0 |
| 3 | 0, 1-i, 2, -i/5 |
| 4 | 1/2, 0, 0, 1 |
Formula Used
We compute the inverse of a nonsingular complex matrix A by Gauss–Jordan elimination on the augmented block [A | I]. Using partial pivoting, we perform row scaling by the pivot and eliminate above and below to reduce the left block to I; the right block becomes A−1.
- Complex division: a/b = \(\frac{(a_r + i a_i)}{(b_r + i b_i)} = \frac{(a_r b_r + a_i b_i) + i(a_i b_r - a_r b_i)}{b_r^2 + b_i^2}\)
- Magnitude: |z| = \(\sqrt{r^2 + i^2}\); pivot chosen by maximal magnitude for stability.
- Verification: compute A · A−1 and measure the maximum absolute deviation from I.
How to Use
- Select the matrix order and enter complex or rational entries.
- Adjust decimal places or switch on Fraction display mode.
- Click Compute Inverse; status and error metrics will appear.
- Toggle the elimination log for detailed row operations and pivots.
- Export the inverse as CSV or PDF; copy to clipboard if needed.
- Use Load Example, Random, or paste CSV to accelerate input.
FAQs
Use forms like a+bi, a-bi, bi, -bi, or real numbers. You can also enter rationals like p/q in real and imaginary parts.
You can choose orders from 2 up to 12. Larger sizes increase time and rounding error; consider higher precision when formatting or use symbolic tools for exact arithmetic.
If the algorithm cannot find a pivot with magnitude above the tolerance, the matrix is treated as singular or ill-conditioned. The status will display an appropriate message.
After inversion, we compute A · A−1 and compare against the identity matrix. The badge shows the maximum absolute entry-wise deviation.
Yes. Use Paste CSV/Text to paste comma or tab-separated values. The grid will fill automatically, trimming extra rows or columns.
Results are formatted as rational numbers for each real and imaginary part using best-fit fractions under the chosen maximum denominator. Computations remain floating-point; formatting is exact only when representable.