Enter Matrix Values
Formula Used
Minor: Mij = determinant of the matrix formed by deleting row i and column j.
Cofactor: Cij = (-1)i+j × Mij
Cofactor Matrix: C = [Cij]
Adjugate: adj(A) = CT
Inverse: A-1 = adj(A) / det(A), when det(A) is not zero.
How to Use This Calculator
- Select a square matrix size from 2 × 2 to 5 × 5.
- Enter each matrix value in the grid.
- Choose the decimal precision for output values.
- Press the calculate button.
- Review minors, cofactors, adjugate, determinant, rank, and inverse status.
- Use the CSV or PDF button to save the result.
Example Data Table
| Matrix Size | Input Matrix | Expected Use |
|---|---|---|
| 2 × 2 | [[4, 7], [2, 6]] | Basic determinant, cofactor, and inverse practice. |
| 3 × 3 | [[2, -1, 3], [0, 4, 5], [7, 2, -6]] | Full minor expansion and adjugate study. |
| 4 × 4 | [[1, 2, 0, 3], [4, 1, 2, 0], [0, 5, 1, 2], [3, 0, 4, 1]] | Advanced symbolic style matrix checking. |
Understanding Matrix Cofactors
What Cofactors Mean
Cofactors help describe the deeper structure of a square matrix. Each cofactor belongs to one entry. It measures how that entry contributes to the determinant. The method uses smaller matrices. These smaller matrices are called minors.
Why the Sign Pattern Matters
A minor alone is not always the final value. A sign must be applied. The sign changes by position. The first position is positive. The next is negative. This checkerboard pattern continues across the matrix. It keeps determinant expansion correct.
Connection With Determinants
Cofactors are often used to expand a determinant along a row or column. This is useful when a matrix has zeros. A row with many zeros reduces work. The calculator shows the full cofactor matrix, so you can inspect every entry. This helps students verify manual work quickly.
Adjugate and Inverse Use
The adjugate matrix is the transpose of the cofactor matrix. It plays a key role in the inverse formula. When the determinant is not zero, the inverse equals the adjugate divided by the determinant. If the determinant is zero, the matrix is singular. Then no standard inverse exists.
Practical Benefits
This calculator is useful for algebra, engineering, physics, economics, and data modeling. It supports square matrices up to 5 × 5. It also shows rank, trace, symmetry, and inverse readiness. The graph highlights cofactor values visually. Large positive or negative values become easier to compare. Export buttons help keep records for homework, reports, and audits.
FAQs
1. What is a cofactor of a matrix?
A cofactor is a signed minor. It is found by deleting one row and one column, calculating the smaller determinant, then applying the sign pattern.
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 positive or negative one based on its position.
3. Can this calculator find the adjugate matrix?
Yes. It finds the cofactor matrix first. Then it transposes that matrix to produce the adjugate matrix.
4. Can cofactors help find the inverse?
Yes. If the determinant is not zero, the inverse equals the adjugate matrix divided by the determinant.
5. Why does a zero determinant matter?
A zero determinant means the matrix is singular. A singular matrix has no standard inverse, even if cofactors can still be calculated.
6. What matrix sizes are supported?
This page supports square matrices from 2 × 2 through 5 × 5. Cofactors are only defined for square matrices.
7. Why is there a graph?
The graph displays cofactor values as a heatmap. It helps compare positive, negative, small, and large values quickly.
8. Can I download the results?
Yes. You can download the calculated matrix data as a CSV file or save a formatted PDF report.