Calculate the inverse of a 2×2 matrix with complex numbers using our accurate Inverse Matrix Calculator.
| Step | Description |
|---|---|
| 1 | Input matrix is [[1+i, 2], [3, 4-i]]. |
| 2 | For a 2×2 matrix [[a, b], [c, d]], the inverse is A⁻¹ = (1/det(A)) * [[d, -b], [-c, a]], where det(A) = a*d – b*c. |
| 3 | Compute det(A) = (1+i)*(4-i) – 2*3 = (5+3i) – 6 = -1 + 3i. |
| 4 | Compute 1/det(A) = 1/(-1+3i). Rationalize: ((-1-3i)/((-1+3i)(-1-3i))) = (-1-3i)/(10) = -0.1 – 0.3i. |
| 5 | Form adjugate matrix: [[4-i, -2], [-3, 1+i]]. |
| 6 | Compute A⁻¹ = (-0.1-0.3i) * [[4-i, -2], [-3, 1+i]] = [[-0.1-0.3i, 0.2], [0.3, 0.1+0.3i]]. |
| Final | Inverse Matrix: [[-0.1-0.3i, 0.2], [0.3, 0.1+0.3i]]. |
Inverse Matrix Calculator – Compute Matrix Inverses Instantly
Struggling with matrix inversion? Use our Inverse Matrix Calculator to compute the inverse of any square matrix in seconds—perfect for students, engineers, and data professionals.
What is an Inverse Matrix Calculator?
An Inverse Matrix Calculator finds the inverse (A⁻¹) of a given square matrix A, such that:
mathematicaCopyEditA × A⁻¹ = I
Where I is the identity matrix.
This tool is essential for solving linear equations, finding transform matrices, performing matrix division, and more.
How to Use the Calculator
- Select matrix size (2×2, 3×3, 4×4, etc.)
- Enter all matrix elements
- Click “Calculate”
Instantly get:
- The inverse matrix
- Optional: original matrix determinant and verification that A × A⁻¹ = I
Why Use This Tool?
| Feature | Benefit |
|---|---|
| 📈 Educational | Helpful for homework, tests, and learning |
| 🛠 Engineer-Ready | Ideal for signal processing, graphics, control systems |
| ⚡ Instant Output | Results in real-time—no waiting |
| 🆓 100% Free & Online | No sign-up, no ads, no limits |
| 📱 Mobile-Friendly | Works seamlessly across all devices |
Who Should Use It?
This tool is perfect for:
- 🧑🎓 Math students tackling linear algebra
- 📊 Data scientists and programmers using matrix operations
- 🔧 Engineers in robotics, 3D graphics, and control systems
- 🧠 Anyone needing quick matrix inversions
FAQs
What if the matrix is singular (determinant = 0)?
No inverse exists for singular matrices. The tool will alert you.
What matrix sizes are supported?
You can compute inverses for 2×2, 3×3, 4×4, or larger matrices (depending on implementation).
Is it really free?
Absolutely—our calculator is always free to use online.