Enter Matrix Values
Fill all nine matrix entries. Then choose a rounding precision and calculate minors, cofactors, adjugate, determinant, inverse, and graph output.
Example Data Table
Use this sample matrix to test the calculator and compare your computed results.
| Example Input Matrix | |||
|---|---|---|---|
| Col 1 | Col 2 | Col 3 | |
| Row 1 | 2 | -1 | 3 |
| Row 2 | 0 | 4 | 5 |
| Row 3 | 1 | -2 | 6 |
| Example Cofactor Matrix | |||
|---|---|---|---|
| Col 1 | Col 2 | Col 3 | |
| Row 1 | 34 | 5 | -4 |
| Row 2 | 0 | 9 | 3 |
| Row 3 | -17 | -10 | 8 |
Example Determinant: 51
Example Adjugate: Transpose of the cofactor matrix.
Formula Used
1. Minor of each entry: Remove the entry’s row and column. Then compute the determinant of the remaining 2×2 matrix.
2. 2×2 determinant: For a 2×2 matrix [[a, b], [c, d]], determinant = ad − bc.
3. Cofactor formula: Cij = (−1)i+j Mij, where Mij is the minor.
4. Sign pattern: [[+, −, +], [−, +, −], [+, −, +]].
5. Adjugate matrix: adj(A) = transpose of the cofactor matrix.
6. Determinant using first row: det(A) = a11C11 + a12C12 + a13C13.
7. Inverse formula: A−1 = adj(A) / det(A), only when det(A) ≠ 0.
How to Use This Calculator
- Enter all nine values of your 3×3 matrix.
- Select the number of decimal places for displayed results.
- Click Calculate Cofactors to process the matrix.
- Review the minor matrix, cofactor matrix, adjugate, determinant, inverse, and the detailed steps table.
- Use the export buttons to save the results as CSV or PDF.
FAQs
1. What does a cofactor represent in a 3×3 matrix?
A cofactor combines a minor with a sign. It helps build the adjugate matrix and is central to determinant expansion and inverse calculations.
2. What is the difference between a minor and a cofactor?
A minor is the determinant of the remaining 2×2 matrix after removing one row and one column. A cofactor applies the alternating sign pattern to that minor.
3. Why is the sign pattern important?
Without the correct sign pattern, the cofactor matrix becomes wrong. That also makes the determinant expansion, adjugate matrix, and inverse matrix incorrect.
4. Can this calculator find the inverse matrix too?
Yes. When the determinant is nonzero, the calculator transposes the cofactor matrix to get the adjugate and divides by the determinant.
5. What happens if the determinant is zero?
A zero determinant means the matrix is singular. The calculator still shows minors, cofactors, and adjugate, but no inverse exists.
6. Why does the result section appear above the form?
This layout shows the answer immediately below the header after submission. It helps users compare results without scrolling past other content first.
7. What does the Plotly heatmap show?
The heatmap visualizes the cofactor matrix. Larger magnitudes stand out, and sign changes become easier to inspect across rows and columns.
8. Is this tool useful for checking manual homework steps?
Yes. The detailed cofactor table shows each 2×2 determinant, sign, and final cofactor. That makes manual verification much easier.