Advanced Matrix Cofactor Calculator
Enter a square matrix. The calculator finds minors, cofactors, adjugate, determinant, and inverse status.
Formula Used
Minor of element aij:
Mij = determinant of the matrix formed after deleting row i and column j.
Cofactor of element aij:
Cij = (-1)i+j × Mij
Cofactor matrix:
C = [Cij]
Adjugate matrix:
adj(A) = transpose of the cofactor matrix.
Inverse condition:
A-1 = adj(A) ÷ det(A), when det(A) ≠ 0.
How to Use This Calculator
- Select the square matrix size from 2 × 2 to 6 × 6.
- Enter each matrix value in its correct row and column.
- Choose the decimal precision for rounded display.
- Press the calculate button to generate results.
- Review minors, cofactors, adjugate, determinant, and inverse status.
- Use CSV or PDF buttons to save the calculation output.
Example Data Table
This sample 3 × 3 matrix is already available through the fill example button.
| Row | Column 1 | Column 2 | Column 3 |
|---|---|---|---|
| Row 1 | 2 | -1 | 3 |
| Row 2 | 0 | 4 | 5 |
| Row 3 | 1 | 2 | -2 |
Understanding Cofactors in Matrix Algebra
What a Cofactor Means
A cofactor is a signed minor. It measures how one matrix entry supports the determinant. First, remove the row and column of the selected entry. Then find the determinant of the remaining smaller matrix. Finally, multiply that minor by the sign pattern. The signs alternate like a checkerboard.
Why Cofactors Matter
Cofactors are important in determinant expansion. They are also used to build the adjugate matrix. The adjugate helps find an inverse when the determinant is not zero. This makes cofactors useful in linear equations, geometry, engineering models, and computer graphics.
Reading the Sign Pattern
The sign is based on the row and column position. Use positive signs when the row number plus column number is even. Use negative signs when it is odd. This rule creates the pattern positive, negative, positive across the first row. The next row starts negative.
Minors and Determinants
A minor is only the determinant of the reduced matrix. It does not include the sign. The cofactor includes both the minor and the sign. For a 3 × 3 matrix, every minor is a 2 × 2 determinant. For larger matrices, the reduced determinant is expanded again.
Adjugate and Inverse
The cofactor matrix becomes the adjugate after transposition. If the determinant of the original matrix is not zero, the inverse can be found by dividing the adjugate by that determinant. If the determinant is zero, the matrix is singular. A singular matrix has no inverse.
Best Use Cases
This calculator is useful for checking classroom work. It is also helpful for verifying long hand calculations. You can compare every cofactor position. The step table shows the sign, minor determinant, and final cofactor. The graph gives a fast visual view of large positive and negative values.
Frequently Asked Questions
1. What is a cofactor matrix?
A cofactor matrix contains every signed minor of a square matrix. Each entry is found by deleting one row and one column, calculating the remaining determinant, and applying the alternating sign rule.
2. What is the difference between a minor and a cofactor?
A minor is the determinant of a reduced matrix. A cofactor is that minor multiplied by a positive or negative sign based on its row and column position.
3. Can this calculator find an adjugate matrix?
Yes. It calculates the cofactor matrix first. Then it transposes that matrix to form the adjugate matrix, which is also called the adjoint matrix.
4. When does the inverse matrix exist?
The inverse exists only when the determinant is not zero. If the determinant equals zero, the matrix is singular and cannot be inverted.
5. What matrix sizes are supported?
This page supports square matrices from 2 × 2 through 6 × 6. Larger matrices need more recursive determinant calculations and may be slower.
6. Why do cofactor signs alternate?
The sign follows (-1) raised to row plus column. This creates a checkerboard pattern of positive and negative signs across the matrix.
7. Can I export the result?
Yes. The page includes CSV and PDF download buttons. These help save the input matrix, cofactor values, and step table for later review.
8. Is decimal input allowed?
Yes. You can enter integers, negative numbers, and decimal values. The selected precision controls how many decimal places appear in the displayed result.