Enter 4×4 Matrix
Determinant: —
Tip: Auto mode expands along the row/column with the most zeros.
Laplace Expansion Steps
Example Data Table
Click Load Example to populate this 4×4 upper‑triangular matrix (determinant is the product of its diagonal entries).
| Row\Col | c1 | c2 | c3 | c4 |
|---|---|---|---|---|
| r1 | 1 | 2 | 0 | 0 |
| r2 | 0 | 3 | 4 | 0 |
| r3 | 0 | 0 | 5 | 6 |
| r4 | 0 | 0 | 0 | 7 |
For this example, the determinant equals 1·3·5·7 = 105.
Formula Used
For a 4×4 matrix \(A=[a_{ij}]\), the Laplace (cofactor) expansion along row \(i\) is \(\det(A)=\sum_{j=1}^{4} (-1)^{i+j}\, a_{ij}\, \det(M_{ij})\), where \(M_{ij}\) is the \(3 \times 3\) minor obtained by deleting row \(i\) and column \(j\). Similarly, expansion can be performed along any column \(j\).
Signs follow the checkerboard pattern \(\begin{smallmatrix}+&-&+&-\\-&+&-&+\\+&-&+&-\\-&+&-&+\end{smallmatrix}\). Each term uses the 3×3 determinant computed via the rule of Sarrus.
How to Use
- Enter numbers in the 4×4 grid. Empty cells count as zero.
- Select an expansion mode: Auto, a specific row, or a specific column.
- Press Compute Determinant to see the full cofactor expansion steps.
- Use Download CSV or Download PDF to export results.
- Try Load Example to explore a structured, easy‑to‑verify case.
FAQs
It expresses a determinant as a sum of products of entries and cofactors, where each cofactor is a signed minor (the determinant of a sub‑matrix with one row and column removed).
Choose a row or column with many zeros to reduce the number of non‑zero terms in the expansion, making calculations faster and numerically more stable.
Each minor is a 3×3 matrix formed by deleting one row and one column. Its determinant is calculated using the rule of Sarrus, a direct formula with six products.
Laplace expansion scales factorially and becomes expensive for large sizes. It is ideal for small matrices or for educational step‑by‑step demonstrations of cofactors and minors.
Yes. The calculator accepts integers, decimals, and negative values. Blank cells count as zero. Computation uses standard floating‑point arithmetic with readable rounding in the display.