Lewis Carroll identity

In linear algebra, the Lewis Carroll identity is an identity involving minors of a square matrix proved by Charles Lutwidge Dodgson (better known by his pseudonym Lewis Carroll), who used it in a method of numerical evaluation of matrix determinants called the Dodgson condensation. From the modern perspective, the Lewis Carroll identity expresses a straightening law in the algebra of polynomial functions of matrices.

Formulation

Let A be an n × n matrix with entries in a commutative ring, and A_ij (i, j = 1, 2) denote its (n − 1) × (n − 1) submatrices of A formed by the (n − 1) first (i = 1) or last (i = 2) rows, and the (n − 1) first (j = 1) or last (j = 2) columns. Let B be the (n − 2) × (n − 2) submatrix of A formed by the rows and columns from 2 to n − 1. The Lewis Carroll identity states that

$(\det A)(\det B)=(\det A_{11})(\det A_{22})-(\det A_{12})(\det A_{21}).\,$

If the matrix B is nonsingular, then dividing by its determinant leads to an expression for the determinant of A in terms of the determinants of orders 1 and 2 lower than the order of A. Recursive application of this procedure is the method of Dodgson condensation.

References

• Bressoud, David (1999), Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture, MAA Spectrum, Washington, D.C.: Mathematical Association of America, ISBN 978-0-521-66646-6.
• Dodgson, C. L. (1866), "Condensation of determinants, being a new and brief method for computing their arithmetical values", Proceedings of the Royal Society of London, 15: 50–55, JSTOR 112607.