the process, with most of the determinants precalculated: Hence, we arrive at a determinant of 36. Desnanot–Jacobi identity and proof of correctness of the condensation algorithm The proof that the condensation method computes the determinant of the matrix if no divisions by zero are encountered is based on an identity known as the Desnanot–Jacobi identity (1841) or, more generally the Sylvester determinant identity (1851). Let be a square matrix, and for each , denote by the matrix that results from by deleting the -th row and the -th column. Similarly, for , denote by the matrix that results from by deleting the -th and -th rows and the -th and -th columns. Desnanot–Jacobi identity Proof of the correctness of Dodgson condensation Rewrite the identity as Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix of order , the matrix in the -th stage of the computation (where the first stage corresponds to the matrix itself) consists of all the connected minors of order of , where a connected minor is the determinant of a connected sub-block of adjacent entries of . In particular, in the last stage , one gets a matrix containing a single element equal to the unique connected minor of order , namely the determinant of . Proof of the Desnanot-Jacobi identity We follow the treatment in Bressoud's book; for an alternative combinatorial proof see the paper by Zeilberger. Denote (up to sign, the -th minor of ), and define a matrix by (Note that the first and last column of are equal to those of the adjugate matrix of ). The identity is now obtained by computing in two ways. First, we can directly compute the matrix product (using simple properties of the adjugate matrix, or alternatively using the formula for the expansion of a matrix determinant in terms of a row or a column) to arrive at where we use to denote the -th entry of . The determinant of this matrix is . Second, this is equal to the product of the determinants, . But clearly so the identity follows from equating the two expressions we obtained for and dividing out by (this is allowed if one thinks of the identities as polynomial identities over the ring of polynomials in the indeterminate variables ). Notes References and further reading Bressoud, David M., Proofs and Confirmations: