Matrix Cofactor Calculator

Calculator

Matrix Size (n×n)

Display Precision

Outputs are rounded for display only.

Show minor matrix

Show cofactor values

Selected Entry Cofactor

Compute one cofactor using (i, j).

1-based

Row i

Col j

Uses Cᵢⱼ = (−1)^(i+j) · det(Mᵢⱼ).

Determinant (always computed)

Adjugate (always computed)

Inverse when possible (always attempted)

Matrix Input

Fill every cell. Decimals and negatives are allowed.

Tip: use Tab to move quickly.

Example Data Table

Sample 3×3 input and a matching cofactor output.

Example Input Matrix A

1	2	3
0	4	5
1	0	6

Determinant for this example is 22.

Example Cofactor Matrix C

24	5	-4
-12	3	2
-2	-5	4

Adjugate is the transpose of C.

Formula Used

Minor: Mᵢⱼ = det(A with row i and column j removed)
Cofactor: Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ
Cofactor Matrix: C = [Cᵢⱼ]
Adjugate: adj(A) = Cᵀ
Inverse (when det(A) ≠ 0): A⁻¹ = adj(A) / det(A)

How to Use This Calculator

Select the matrix size and precision.
Enter all matrix values in the grid.
Optionally enable minor matrix and cofactor values.
Press Submit to see results above the form.
Use CSV or PDF buttons to download your latest results.

FAQs

1) What is a cofactor in a matrix?

A cofactor is a signed minor. Remove row i and column j, take the determinant of the remaining matrix (the minor), then multiply by (−1)^(i+j).

2) What is the difference between a minor and a cofactor?

The minor is only the determinant of the reduced matrix. The cofactor applies a sign pattern to that minor using (−1)^(i+j).

3) What is the cofactor matrix used for?

The cofactor matrix helps build the adjugate. The adjugate is used in the classical inverse formula: A⁻¹ = adj(A)/det(A) when the determinant is not zero.

4) When does a matrix have an inverse?

A square matrix has an inverse only when its determinant is nonzero. If the determinant is zero, the matrix is singular and cannot be inverted.

5) Why do I see “no inverse available” sometimes?

That message appears when the determinant is zero or extremely close to zero. In that case, dividing by the determinant would be unstable or undefined.

6) What sizes are supported here?

This tool supports sizes from 2×2 up to 6×6. Larger sizes can be added, but cofactor-based operations become computationally heavier as n grows.

7) Are the results exact or rounded?

Internally the calculator uses floating-point arithmetic. The precision setting controls how many decimals are displayed and exported, but calculations still use full float values.

8) What do the CSV and PDF downloads include?

Downloads include the chosen matrix output (cofactor, adjugate, or inverse). The PDF also includes the matrix size and determinant to keep your report self-contained.