Cauchy Binet Formula Preview

Math Linear Algebra • Determinants and Rank

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

Preview the Cauchy–Binet formula for \(A\in\mathbb{R}^{m\times n}\) and \\(B\in\mathbb{R}^{n\times m}\\) (\\(m\le n\\)): \[ \det(AB)=\sum_{S\subseteq\{1,\dots,n\},\,|S|=m} \\det(A_{:,S})\\,\\det(B_{S,:}). \] Click a subset \\(S\\) to highlight the corresponding columns of \\(A\\) and rows of \\(B\\).

m (rows of A, cols of B)

n (cols of A, rows of B)

Zero tolerance

Display decimals

Sort subsets

Top-k (list)

Steps

Hide zero terms Show \(AB\) and \(\det(AB)\) check Show only top-k by \(|\text{term}|\) in list

Entries accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. Dimensions are linked: \(A\) is \(m\times n\), \(B\) is \(n\times m\), so \(AB\) is \(m\times m\).

Matrix \(A\) (m×n)

2×3

Selected subset columns of \(A\) will highlight after Calculate / selection.

Matrix \(B\) (n×m)

3×2

Selected subset rows of \(B\) will highlight after Calculate / selection.

Results

Number of subsets \(\binom{n}{m}\)

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Cauchy–Binet sum

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\(\det(AB)\)

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Difference \(|\det(AB)-\sum_S|\)

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Ready

Minor highlight

Left: \(A\) with selected columns \(S\). Right: \(B\) with selected rows \(S\). Bottom: the minors \(A_{:,S}\), \(B_{S,:}\).

Click a subset in the list to highlight its minors • Drag to pan • wheel to zoom • double-click to reset view

Subset terms

Each term is \(\det(A_{:,S})\det(B_{S,:})\) for a subset \(S\subseteq\{1,\dots,n\}\) with \(|S|=m\).

Enter matrices and click “Calculate”.

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Theory

What the Cauchy–Binet formula says

The usual identity \(\det(AB)=\det(A)\det(B)\) requires square matrices. The Cauchy–Binet formula extends determinant-style reasoning to the non-square product \(A\in\mathbb{R}^{m\times n}\) and \(B\in\mathbb{R}^{n\times m}\) (with \(m\le n\)), so that \(AB\) is square: \[ \det(AB)=\sum_{S\subseteq\{1,\dots,n\},\,|S|=m}\det(A_{:,S})\det(B_{S,:}). \] Here \(A_{:,S}\) is the \(m\times m\) submatrix of \(A\) formed by choosing columns indexed by \(S\), and \(B_{S,:}\) is the \(m\times m\) submatrix of \(B\) formed by choosing rows indexed by \(S\).

What are “minors” in this context?

A minor is a determinant of a square submatrix. Because \(A\) is \(m\times n\) with \(m\le n\), it has many \(m\times m\) column-minors \(\det(A_{:,S})\) (one for each subset \(S\)). Similarly, \(B\) has \(m\times m\) row-minors \(\det(B_{S,:})\).

The Cauchy–Binet formula says that \(\det(AB)\) is a weighted sum of products of these minors. If many minors are zero (for example when matrices are sparse or have dependent columns/rows), only a few subsets contribute.

Why it works (intuition)

One common proof viewpoint uses multilinearity of the determinant in the columns of a matrix. The columns of \(AB\) are linear combinations of the columns of \(A\), with coefficients coming from \(B\). When you expand \(\det(AB)\) using multilinearity and alternation, only terms selecting \(m\) distinct columns survive, and those surviving terms group into minors indexed by subsets \(S\).

Another interpretation uses exterior algebra (wedge products): determinants measure oriented \(m\)-dimensional volume, and Cauchy–Binet is the coordinate expression of how these volumes transform under the composition represented by \(A\) and \(B\).

Small worked shape: 2×3 times 3×2

If \(m=2\) and \(n=3\), there are \(\binom{3}{2}=3\) subsets: \[ S\in\{\{1,2\},\{1,3\},\{2,3\}\}. \] The formula becomes a sum of three products: \[ \det(AB)=\det(A_{:,12})\det(B_{12,:})+\det(A_{:,13})\det(B_{13,:})+\det(A_{:,23})\det(B_{23,:}). \] The calculator lists these terms explicitly and lets you click each subset to highlight the corresponding minors.

University note: Grassmannians and Plücker coordinates

The minors \(\det(A_{:,S})\) are closely related to Plücker coordinates of an \(m\)-dimensional subspace of \(\mathbb{R}^n\), and the set of all such subspaces forms the Grassmannian. In this language, Cauchy–Binet describes how these coordinates combine under composition of linear maps.

For numerical computation with larger matrices, direct minor summation is usually not used, but it remains a powerful theoretical identity.

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