Enter a 3x3 Matrix
Use any integers or decimals. Then generate the full cofactor matrix, determinant, minor matrix, and adjugate.
Example Matrix Cases
| Case | Input Matrix | Cofactor Matrix | Determinant |
|---|---|---|---|
| Example 1 | [2, -1, 3] [0, 4, 5] [1, -2, 6] |
[34, 5, -4] [0, 9, 3] [-17, -10, 8] |
51 |
| Example 2 | [1, 2, 3] [4, 5, 6] [7, 8, 9] |
[-3, 6, -3] [6, -12, 6] [-3, 6, -3] |
0 |
| Example 3 | [3, 1, 2] [2, 0, -1] [4, 5, 2] |
[5, -8, 10] [8, -2, -11] [-1, 7, -2] |
27 |
How the Cofactor Matrix Is Calculated
For each entry, remove its row and column. Compute the remaining 2x2 determinant. That gives the minor value.
Then apply the sign pattern below:
| + | - | + |
| - | + | - |
| + | - | + |
Cofactor formula: Cij = (-1)i+j × Mij
2x2 determinant formula: ad - bc
3x3 determinant formula: a11C11 + a12C12 + a13C13
Using This Calculator
- Enter all nine matrix values into the input fields.
- Click Calculate Cofactor Matrix.
- Review the determinant, minor matrix, and cofactor matrix.
- Check the adjugate matrix for inverse-related work.
- Use the heatmap to inspect relative cofactor magnitudes.
- Download the result as CSV or PDF.
Frequently Asked Questions
1. What is a cofactor matrix?
A cofactor matrix contains signed minors for every matrix position. Each value comes from a 2x2 determinant after removing one row and one column.
2. What is the difference between a minor and a cofactor?
A minor is the determinant of the reduced 2x2 matrix. A cofactor is that minor multiplied by a sign from the alternating sign pattern.
3. Why does the sign pattern alternate?
The alternating pattern ensures determinant expansion works correctly. It follows the rule (-1)i+j for each row and column position.
4. Can this calculator help find an inverse?
Yes. First find the cofactor matrix. Then transpose it to get the adjugate. Divide that matrix by the determinant when the determinant is nonzero.
5. What happens if the determinant is zero?
A zero determinant means the matrix is singular. It has no inverse, even though the calculator can still show its minors and cofactors.
6. Does the calculator support decimals?
Yes. You can enter integers, negatives, and decimal values. The calculator handles them directly and formats the result cleanly.
7. What does the heatmap show?
The heatmap displays the cofactor matrix visually. Larger magnitudes stand out, while negative and positive values become easier to compare quickly.
8. Is this calculator only for 3x3 matrices?
Yes. This page is designed for 3x3 matrices only. It uses fixed inputs and formulas built specifically for that matrix size.