Determinant Using Laplace Expansion Calculator

Create square matrices, choose expansions, and verify determinants. See signed cofactors, recursive minors, and summaries. Download clean reports, examples, and visual matrix insights fast.

Calculator form

Build your square matrix

This page keeps a single-column layout, while the form itself shifts into 3, 2, and 1 columns across large, medium, and mobile screens.

Leave any box blank to treat it as zero.

Matrix size

Expand along

Expansion index

Displayed recursion depth

Decimal precision

Matrix entries

Formula used

Laplace expansion formula

Row expansion: det(A) = Σ a_ijC_ij, taken across a fixed row.

Column expansion: det(A) = Σ a_ijC_ij, taken down a fixed column.

Cofactor: C_ij = (−1)^i+j det(M_ij), where M_ij is the minor formed after deleting row i and column j.

The calculator respects your chosen top-level row or column, then recursively evaluates minors. It also suggests a faster branch with more zeros because zero terms disappear instantly.

How to use this calculator

Quick usage guide

Choose a matrix size from 2 × 2 through 6 × 6.
Pick whether the first expansion should use a row or a column.
Enter the index number and type each matrix entry. Fractions are allowed.
Select how many recursive levels you want printed in the walkthrough.
Press calculate to place the determinant, graph, and steps above the form.

Example data table

Worked example

This example uses row 1 for the first Laplace expansion. The determinant equals 17.

Example	a11	a12	a13	a21	a22	a23	a31	a32	a33	Determinant
Matrix A	2	1	3	0	-1	4	5	2	1	17

FAQs

Frequently asked questions

1. What is Laplace expansion?

Laplace expansion computes a determinant by selecting one row or column, attaching alternating signs, and multiplying each entry by the determinant of its minor matrix.

2. Which row or column should I choose?

Choose the row or column containing the most zeros. Zero entries eliminate entire branches, so the expansion becomes shorter and easier to verify.

3. Can I enter fractions instead of decimals?

Yes. Inputs like 1/2, -3/4, and 7/5 are accepted. The calculator converts them into decimals before evaluating the determinant and all minors.

4. Why do the cofactor signs alternate?

The sign pattern comes from (−1)^(i+j). It creates the familiar checkerboard of plus and minus signs required for valid cofactor expansion.

5. Why can a large matrix take longer?

Laplace expansion is recursive. Each nonzero term creates a smaller determinant, so work grows quickly with size. Matrices with many zeros are much faster.

6. Does the chosen row or column change the answer?

No. Any valid row or column gives the same determinant. Only the amount of work changes, which is why a zero-rich line is usually best.

7. What does the Plotly graph show?

The graph plots top-level cofactor contributions from the selected expansion. Positive and negative bars make it easier to see which terms dominate the final determinant.

8. What do the export buttons include?

The CSV button downloads matrix values and contribution data. The PDF button captures the visible report section so you can save or share the worked solution.

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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.