Laplace Expansion Tool

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Compute \(\det(A)\) using Laplace (cofactor) expansion along a chosen row or column: \[ \det(A)=\sum_j a_{rj}C_{rj}\quad\text{or}\quad \det(A)=\sum_i a_{ic}C_{ic}, \qquad C_{ij}=(-1)^{i+j}\det(M_{ij}). \] Use Auto-best to expand along the row/column with the most zeros.

Size (n×n)

Expand along

Computation mode

Recursive minors

Zero tolerance

Display decimals

Steps

Tree depth

Show cofactor sign grid \((-1)^{i+j}\)

Tip: Auto-best chooses the row/column with the most entries close to zero (within tolerance) to reduce work. Entries accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Matrix \(A\)

3×3

Selected expansion row/column will be highlighted after you click “Calculate”.

Cofactor sign grid \(\;(-1)^{i+j}\;\)

Results

\(\det(A)\)

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Expansion used

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Nonzero terms used

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Skipped zero-terms

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Ready

Expansion path tree

Shows the top-level terms in the Laplace expansion. Depth 2 expands minors only for small matrices.

Drag to pan • wheel to zoom • double-click to reset view

Enter a matrix and click “Calculate”.

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Theory

Laplace (cofactor) expansion

The Laplace expansion expresses a determinant as a sum of cofactors along a chosen row or column. For a row expansion along row \(r\), \[ \det(A)=\sum_{j=1}^{n} a_{rj}C_{rj}. \] For a column expansion along column \(c\), \[ \det(A)=\sum_{i=1}^{n} a_{ic}C_{ic}. \] Each cofactor is \[ C_{ij}=(-1)^{i+j}\det(M_{ij}), \] where \(M_{ij}\) is the \((n-1)\times(n-1)\) matrix obtained by deleting row \(i\) and column \(j\).

Checkerboard sign grid \((-1)^{i+j}\)

Cofactors use the alternating sign \((-1)^{i+j}\), which forms a checkerboard:

\[ \begin{bmatrix} + & - & + & - & \cdots\\ - & + & - & + & \cdots\\ + & - & + & - & \cdots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} \]

If \(i+j\) is even, the sign is \(+\). If \(i+j\) is odd, the sign is \(−\). The calculator can display this sign grid and highlight the row/column you expand along, which is useful for hand computation.

Minors and cofactors

The minor matrix \(M_{ij}\) is formed by deleting row \(i\) and column \(j\). Its determinant \(\det(M_{ij})\) is the minor determinant. Multiplying by the checkerboard sign produces the cofactor \(C_{ij}\).

Which row/column should you expand?

Laplace expansion can be extremely fast when you expand along a row/column with many zeros, because most terms vanish. A good manual strategy is to choose the row or column with the most zeros (or simplest numbers).

The calculator’s Auto-best option automates this by selecting the row/column with the most entries close to zero (within a chosen tolerance).

Pure Laplace vs hybrid computation

A fully recursive Laplace expansion grows quickly for dense matrices. To keep the tool responsive, it offers a hybrid mode: it shows the top-level Laplace expansion, but computes each minor determinant using elimination.

University note: efficiency and structure

For large matrices, determinants are usually computed using factorization methods (LU/QR), often exploiting sparsity or block structure. Laplace expansion remains valuable for proofs, symbolic work, and sparse matrices where many terms become zero.

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