Enter any square matrix and choose expansion row. Get clear minors, cofactors, and determinant totals. Practice faster, export proofs, and improve linear algebra skills.
For an n×n matrix A, expanding along a chosen row i: det(A) = Σj=1..n aij Cij, where Cij = (−1)i+j det(Mij) and Mij is the minor formed by deleting row i and column j.
Expanding along a column j follows the same idea, summing over i. Base cases use direct formulas for 1×1 and 2×2 determinants.
Example 3×3 matrix (a sparse row improves expansion speed):
| Row | Values | Zeros count | Suggested expansion |
|---|---|---|---|
| 1 | [2, 0, 1] | 1 | Row 2 is better |
| 2 | [3, 0, 0] | 2 | Row 2 |
| 3 | [5, 1, 1] | 0 | Not preferred |
Cofactor expansion, also called Laplace expansion, expresses a determinant as a signed sum of smaller determinants. Each term uses an entry aij multiplied by its cofactor Cij. This structure is useful for teaching, proofs, and manual work when a row or column contains many zeros. It also clarifies why swapping two rows flips the sign of the result. The method links geometric volume scaling to algebraic recursion in a memorable way.
The number of recursive minors grows quickly with matrix size. A 5×5 determinant can require dozens of 4×4 and 3×3 evaluations if you expand across a dense row. Expanding across a sparse row or column reduces terms, because every zero entry contributes nothing. When entries repeat or share factors, selecting that line simplifies arithmetic and reduces mistakes.
For an element aij, the minor Mij is the determinant of the matrix formed by deleting row i and column j. The cofactor is Cij = (−1)i+j Mij. The alternating checkerboard sign pattern is essential and is shown in the steps. Many learners start with a plus in the top left, then alternate across and down.
This calculator prints intermediate minors and the final sum, letting learners verify each stage against notes. Exporting a CSV summary preserves the chosen expansion path, determinant value, and key terms. The PDF output creates a shareable record for tutoring and revision. Because each minor is visible, it is easier to spot an incorrect deleted row or column, a common source of errors during exam practice.
Cofactor expansion has factorial-like growth in operations, so it is not preferred for large numeric problems. In applied settings, row reduction or LU decomposition is usually faster and more stable for floating point data. Still, controlled cofactor expansion remains valuable for symbolic and patterned matrices, and for demonstrating linear algebra properties. For speed, create zeros with simple operations before expanding. If entries are integers, exact arithmetic avoids rounding drift and improves checking.
A cofactor is a signed minor. You delete one row and one column to form a smaller determinant, then multiply by (−1)i+j to apply the correct sign.
Pick the row or column with the most zeros or simple numbers. Fewer nonzero terms means fewer minors to compute, which saves time and reduces arithmetic errors.
The alternating signs come from the factor (−1)i+j. This pattern ensures the expansion matches determinant properties, including sign changes under row swaps and linearity in each row.
It becomes slow as size increases because the number of minors grows rapidly. For larger numeric matrices, methods like row reduction or LU decomposition are typically much faster.
Yes. Enter negative values or decimals in any cell. The output rounds display values for readability while keeping internal calculations consistent for typical classroom and practice use.
Exports include the matrix size, chosen expansion mode, determinant value, and the matrix itself. They are designed to help you share results, store evidence, or cross-check work later.
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.