4x4 Matrix Inverse Calculator

Matrix A (4×4)

You can type decimals, integers, or fractions like -3/4.

Options

Output mode

Decimal precision

Pivot tolerance

Show detailed elimination steps

Tip

If your matrix has large or tiny values, increase precision and keep a small tolerance.

Actions

What you’ll get

Inverse matrix (if it exists)
Determinant and identity-check error
Optional step-by-step row operations
CSV and PDF exports

Example data table

Example input matrix A

1	2	3	4
0	1	4	2
5	2	1	0
2	3	0	1

Determinant: -98

Example inverse A^-1

13/98	-8/49	25/98	-10/49
-12/49	11/49	-8/49	26/49
-17/98	18/49	5/98	-2/49
23/49	-17/49	-1/49	-9/49

Fractions shown for clarity; decimals also supported.

Formula used

A square matrix is invertible only when its determinant is nonzero: det(A) ≠ 0.

One classic relationship is: A^-1 = adj(A) / det(A). For a 4×4 matrix, computing adj(A) directly is lengthy, so this calculator uses Gauss–Jordan elimination.

Gauss–Jordan works by augmenting the matrix: [A | I] and applying row operations until it becomes [I | A^-1]. Partial pivoting improves numerical stability.

How to use this calculator

Enter all 16 values for matrix A, using decimals or fractions.
Choose output mode and decimal precision if needed.
Click Compute inverse to see results above the form.
Use the download buttons to export CSV or PDF.
If you enable steps, scroll through the row-reduction snapshots.

FAQs

1) What matrices can be inverted?

Only 4×4 square matrices with a nonzero determinant are invertible. If det(A) equals zero, no matrix exists that can multiply A to produce the identity matrix.

2) Why do I get “near-singular” warnings?

If the determinant is extremely small, the matrix behaves like singular in floating-point arithmetic. Tiny changes in inputs can cause huge changes in the inverse. Try higher precision or recheck your values.

3) Can I enter fractions like 7/3?

Yes. Type fractions as numerator/denominator, including negatives. The calculator converts them to numeric values, then computes the inverse normally.

4) What does “max identity error” mean?

After computing A^-1, the calculator multiplies A·A^-1 and compares it to the identity matrix. The largest absolute difference is reported as the max identity error.

5) How is the determinant calculated?

The determinant is computed using LU-style elimination with partial pivoting. This avoids the heavy expansion-by-minors approach and generally behaves better for real-world numeric inputs.

6) Why offer a pivot tolerance setting?

Tolerance controls when a pivot is treated as “too small,” which affects stability. A tiny pivot can amplify rounding errors. Keep it small for clean inputs, or raise it slightly for noisy data.

7) What are the detailed steps showing?

They show snapshots of the augmented matrix [A | I] after each key row operation: swaps, scaling a pivot row, and eliminating entries above and below pivots.

8) Why do CSV and PDF downloads use the latest result?

Downloads are generated from the most recent successful calculation stored in your session. If you refresh or open the page in a new session, compute again to enable downloads.