Enter Matrix Values
Provide the nine matrix entries below. Empty inputs are treated as zero, which helps during quick testing.
Example Data Table
Use this sample matrix to verify the calculator output and understand how minors align with matrix positions.
| Example | a11 | a12 | a13 | a21 | a22 | a23 | a31 | a32 | a33 | M11 | M22 | Determinant |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sample A | 2 | 1 | 3 | 0 | 4 | 5 | 7 | 2 | 6 | 14 | -9 | 33 |
Formula Used
Minor definition: The minor of entry aij equals the determinant of the 2×2 submatrix left after deleting row i and column j.
2×2 determinant: |a b; c d| = ad − bc
Example: For M11, remove row 1 and column 1. Then calculate:
M11 = det [ [a22, a23], [a32, a33] ] = a22a33 − a23a32
Cofactor relation: Cij = (−1)i+j Mij
Determinant expansion: det(A) = a11C11 + a12C12 + a13C13
How to Use This Calculator
- Enter the nine values of your 3x3 matrix into the labeled fields.
- Use decimals or negative numbers whenever your problem requires them.
- Click Calculate Minors to generate all nine minors instantly.
- Review the cofactor matrix and determinant for added matrix insight.
- Check the Plotly heatmap to compare minor magnitudes visually.
- Use the CSV button to save structured output for spreadsheets.
- Use the PDF button to export a clean summary for printing.
FAQs
1. What does a matrix minor represent?
A matrix minor is the determinant of the smaller submatrix left after removing one row and one column from the original matrix. It helps reveal local structure around a chosen entry and supports cofactor expansion, adjugate construction, and inverse calculations.
2. Why is this calculator limited to 3x3 matrices?
This tool focuses on one common classroom and exam format. A 3x3 matrix gives nine distinct minors, enough detail for learning patterns without making the interface bulky or slowing down quick problem checks.
3. What is the difference between a minor and a cofactor?
A minor is the raw 2x2 determinant after deleting a row and column. A cofactor applies a sign pattern to that minor using alternating plus and minus positions, shown by the factor (−1)i+j.
4. Can the determinant be found from these minors?
Yes. The determinant of a 3x3 matrix can be expanded from any row or column using cofactors. This calculator shows minors and cofactors together, making determinant verification faster and easier to understand.
5. What happens when the determinant is zero?
A zero determinant means the matrix is singular. It does not have a standard inverse, and its rows or columns are linearly dependent. The minor pattern can still provide valuable structural information.
6. Can I use decimals or negative values?
Yes. The input fields accept decimal and negative numbers, so the calculator works for integer matrices, fractional values, and applied problems where measurements or coefficients are not whole numbers.
7. Why is a heatmap useful here?
The heatmap helps you compare the position and magnitude of minors at a glance. Strong positive or negative values stand out visually, which can reveal symmetry, dominance, or imbalance within the matrix.
8. When would I export results as CSV or PDF?
CSV is useful for spreadsheet analysis, recordkeeping, and homework tables. PDF is useful for sharing polished results, printing problem sets, or keeping a formatted summary of your matrix work.