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The adjoint of a matrix is the transpose of its cofactor matrix.

The adjoint of a matrix A, denoted by adj(A), is defined as the transpose of the cofactor matrix of A. The cofactor matrix of A, denoted by C, is obtained by taking the matrix of cofactors of A and transposing it. The matrix of cofactors of A, denoted by Cof(A), is obtained by taking the matrix of minors of A and multiplying each entry by (-1)^(i+j), where i and j are the row and column indices of the entry, respectively.

In other words, if A = [a_ij] is an n x n matrix, then the (i,j)-entry of Cof(A) is given by (-1)^(i+j) times the determinant of the (n-1) x (n-1) matrix obtained by deleting the i-th row and j-th column of A. Then, the (i,j)-entry of C is given by the (j,i)-entry of Cof(A). Finally, the (i,j)-entry of adj(A) is given by the (j,i)-entry of C.

The adjoint of a matrix is useful in many areas of mathematics, including linear algebra, differential equations, and calculus. One important application is in finding the inverse of a matrix. Specifically, if A is an invertible matrix, then its inverse is given by A^(-1) = (1/det(A)) adj(A), where det(A) is the determinant of A. This formula can be used to find the inverse of a matrix by first computing its adjoint and determinant.

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