Calculate matrix minors with clear steps and validation. Pick a position, or compute all minors. Export tables to CSV or PDF for sharing instantly.
Sample 3x3 matrix and one minor computation preview.
| Matrix A | ||
|---|---|---|
| 2 | -1 | 3 |
| 0 | 4 | 5 |
| 1 | 2 | -2 |
| 0 | 5 |
| 1 | -2 |
For a square matrix A, the minor of entry aij is:
Mij = det(A(i,j))
A(i,j) is the reduced matrix formed by deleting row i and column j. The determinant is computed by standard expansion rules.
In linear algebra, a minor is the determinant of a reduced matrix formed by deleting one row and one column. Minors power cofactor expansion, adjugate construction, and inverse computation for non‑singular matrices. This calculator automates the reduction and determinant steps, so you can focus on reasoning, sign patterns, and documenting each intermediate matrix cleanly.
As matrix size grows, the operation count rises quickly. For an n×n matrix, a single minor requires a determinant of size (n−1)×(n−1). Computing all minors requires n² such determinants, so work scales steeply beyond 5×5. In typical coursework, 2×2 through 4×4 are instant, 5×5 is still comfortable, and 6×6 to 7×7 can feel heavier on older laptops.
For a chosen position (i, j), the tool builds A(i,j) by skipping row i and column j, then evaluates det(A(i,j)). The determinant is computed by expansion with alternating signs, mirroring the textbook rule. Results show the original matrix, the reduced submatrix, and the final minor value. When you switch to “all minors,” the output becomes an n×n table of Mij values ready for cofactor or adjugate work.
Simple checks help validate outputs. If two rows of A(i,j) are identical, the minor must be zero. If a reduced submatrix has a row of zeros, the determinant collapses and the minor is often zero as well. For diagonal or triangular matrices, many minors equal products of remaining diagonal entries, so you can verify several positions quickly. Row swaps flip determinant signs, so pay attention when comparing hand work.
Minors appear in sensitivity analysis, coordinate transforms, and symbolic manipulation, especially when forming cofactors and adjugates. Engineers use them when solving small systems by hand, and analysts meet them in determinant‑based identities and matrix diagnostics. Exporting CSV or PDF lets you attach computations to lab notebooks, reports, or peer review, including the exact reduced matrices used. That traceability matters when results must be reproduced or audited later. For teaching, compare minors across positions to see structure, then use the minor matrix to build cofactors systematically without extra algebra.
A matrix minor Mij is the determinant of the submatrix obtained by deleting row i and column j from the original square matrix.
A cofactor Cij equals (−1)i+j times the minor Mij. Cofactors include an alternating sign pattern; minors are unsigned determinants.
It supports square matrices from 2x2 up to 7x7. Larger sizes are possible in theory, but computation time grows quickly for all-minors mode.
If the reduced submatrix has dependent rows or a full zero row/column, its determinant is zero. This commonly happens with repeated rows or sparse matrices.
Yes. Enter integers, decimals, or scientific notation such as 1.2e-3. Non-numeric entries are flagged and treated as zero for safety.
CSV is convenient for spreadsheets and further analysis. PDF is useful for sharing fixed reports, including the input matrix and the computed minor tables.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.