Enter Matrix Values
Use formats like 3+2i, -4i, 5, or 1-i. The page uses a single-column flow, while the calculator fields adapt to large, medium, and mobile screens.
Plotly Graph
The graph places the determinant on the complex plane. The horizontal axis is the real part, and the vertical axis is the imaginary part.
Example Data Table
| Example | Matrix | Determinant | Use Case |
|---|---|---|---|
| 2 × 2 Complex |
[[2+i, 3-2i], [1+4i, -2+i]] |
-16 - 10i | Quick checks for algebra and introductory linear transformations. |
| 3 × 3 Complex |
[[1+i, 2, 0], [0, 3-i, 1], [2i, 0, 2]] |
8 + 8i | More advanced work in proofs, systems, and signal models. |
Formula Used
For a 2 × 2 complex matrix
det(A) = ad − bc
For a 3 × 3 complex matrix
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Complex result measures
If z = x + yi, then the calculator also shows:
Modulus: |z| = √(x² + y²)
Argument: arg(z) = atan2(y, x)
Conjugate: z̄ = x − yi
Interpretation rule
A matrix is singular when its determinant equals zero. If the determinant is not zero, the matrix is invertible and non-singular.
How to Use This Calculator
1. Select either a 2 × 2 or 3 × 3 matrix.
2. Enter each matrix term using real or complex notation.
3. Click Calculate Determinant to process the matrix.
4. Review the determinant, modulus, angle, conjugate, and matrix status above the form.
5. Use the CSV and PDF buttons to export your computed result.
Frequently Asked Questions
1) What is a complex determinant?
A complex determinant is the determinant of a matrix whose entries may contain real and imaginary parts. It shows whether the matrix is singular and summarizes scaling and orientation effects in complex-valued transformations.
2) Does a zero determinant always mean the matrix is singular?
Yes. When the determinant is zero, the matrix has no inverse. That means the matrix is singular, and its rows or columns are linearly dependent.
3) Which input formats are accepted here?
You can enter values like 5, -3, 2+4i, 7-2i, i, -i, and 3i. Spaces are ignored. The calculator also accepts j by converting it to i automatically.
4) Can I calculate both 2 × 2 and 3 × 3 determinants?
Yes. Choose the matrix size from the selector. The page will show only the fields needed for that size and ignore hidden entries.
5) Why are modulus and argument included?
The determinant is a complex number. Modulus shows its size, while argument shows its angle on the complex plane. Together, they give a stronger interpretation than rectangular form alone.
6) What does the graph show?
The graph places the determinant on an Argand-style plane. The x-axis is the real part, and the y-axis is the imaginary part. This helps you visualize direction and magnitude.
7) Are the CSV and PDF exports useful for reports?
Yes. CSV works well for spreadsheets and data logs. PDF works well for saved notes, classwork, printouts, and quick reporting.
8) What happens if I type an invalid complex value?
The calculator shows an error message and keeps your entered values. You can correct the invalid field and submit again without rebuilding the whole matrix.