Calculator Input
Enter each matrix element as separate real and imaginary parts.
Example Data Table
| a11 | a12 | a21 | a22 | a11 × a22 | a12 × a21 | det(A) |
|---|---|---|---|---|---|---|
| 2 + 3i | 1 − 2i | −4 + i | 3 + 5i | −9 + 19i | −2 + 9i | −7 + 10i |
| 4 + i | −3 + 2i | 5 − i | 2 + 4i | 4 + 18i | −13 + 13i | 17 + 5i |
| 1 + 0i | 2 + i | 3 − i | 4 + 2i | 4 + 2i | 7 + i | −3 + i |
Formula Used
A = [[a11, a12], [a21, a22]]
det(A) = (a11 × a22) − (a12 × a21)
(x + yi)(u + vi) = (xu − yv) + (xv + yu)i
For det(A) = p + qi, modulus = √(p² + q²), and argument = atan2(q, p).
The matrix is invertible when det(A) ≠ 0.
How to Use This Calculator
- Enter the real and imaginary parts for a11, a12, a21, and a22.
- Click Calculate Determinant to process the matrix.
- Read the result block shown below the header and above the form.
- Review the diagonal products, determinant modulus, phase angle, and invertibility status.
- Inspect the Plotly graph to see all important complex values on the Argand plane.
- Use the export buttons to download the result summary as CSV or PDF.
Frequently Asked Questions
1) What does the determinant represent for a complex matrix?
It measures how the matrix scales and rotates complex space. A zero determinant means the transformation collapses dimension and the matrix becomes noninvertible.
2) Is the determinant formula different for complex entries?
No. The structure stays the same: det(A) = a11a22 − a12a21. Only the multiplication and subtraction use complex arithmetic.
3) Why are real and imaginary parts entered separately?
Separate fields reduce parsing errors and make the calculation more reliable. They also allow cleaner validation, plotting, and formatting of results.
4) What does the modulus of the determinant show?
The modulus gives the overall size of the complex determinant. It is the distance of the determinant point from the origin on the complex plane.
5) What does the argument of the determinant mean?
The argument is the angle of the determinant in the complex plane. It helps describe the rotational component of the resulting complex value.
6) When is a 2x2 complex matrix invertible?
A 2x2 complex matrix is invertible exactly when its determinant is not zero. This page checks that condition automatically after each submission.
7) Why does the calculator show diagonal products separately?
Showing both products makes the determinant process transparent. You can verify the major multiplication terms before their final subtraction.
8) What does the Plotly graph display?
It plots the matrix entries, both diagonal products, and the determinant as points on the complex plane. This gives a quick visual understanding of their relationships.