Cofactor and Minor Tool

Math Linear Algebra • Determinants and Rank

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

Compute the minor \(M_{ij}\) (determinant of the submatrix obtained by deleting row \(i\) and column \(j\)) and the cofactor \(C_{ij}=(-1)^{i+j}M_{ij}\). Use the checkerboard sign to verify the cofactor sign.

Size (n×n)

Row index (i)

Column index (j)

Minor det method

Display decimals

Zero tolerance

Steps

Tip: click any matrix cell to set \((i,j)\). Entries accept -3.5, 2e-4, fractions like 7/3, and pi, e.

Matrix \(A\)

3×3

Deleted row \(i\) and column \(j\) are shaded. Remaining entries form the submatrix for the minor.

Minor / Cofactor

Minor \(M_{ij}\)

—

Sign \((-1)^{i+j}\)

—

Cofactor \(C_{ij}\)

—

Expansion term \(a_{ij}C_{ij}\)

—

Cofactor sign checkerboard

Ready

Submatrix shading

Selected cell is red; deleted row/column are gray; remaining cells are used for the minor determinant.

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

Enter a matrix, choose \((i,j)\), then click “Calculate”.

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Theory

Minor and cofactor (definitions)

Let \(A\in\mathbb{R}^{n\times n}\). For an entry \(a_{ij}\), the minor \(M_{ij}\) is the determinant of the \((n-1)\times(n-1)\) submatrix obtained by deleting row \(i\) and column \(j\): \[ M_{ij}=\det(B),\quad B=\text{delete row }i\text{ and column }j\text{ from }A. \]

The cofactor adds the checkerboard sign: \[ C_{ij}=(-1)^{i+j}M_{ij}. \]

Checkerboard sign pattern

The signs \((-1)^{i+j}\) alternate in a checkerboard:

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

If \(i+j\) is even, the sign is \(+1\). If \(i+j\) is odd, the sign is \(-1\).

Why cofactors matter (expansion step)

Cofactors are used in determinant expansion (Laplace expansion). Expanding along row \(i\): \[ \det(A)=\sum_{j=1}^{n} a_{ij}C_{ij}. \]

Each position contributes exactly the expansion term \(a_{ij}C_{ij}\). The tool reports this term for the selected \((i,j)\), which is useful for building a hand-written expansion step.

Higher minors (university note)

The term “minor” is also used more generally for determinants of larger submatrices (not just \((n-1)\times(n-1)\)). Those appear in topics like rank tests via minors and advanced multilinear algebra.

With floating-point data, tiny values may be treated as zero using a tolerance to avoid misleading signs from roundoff.

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