Let M be a square matrix of size n. The (i,j) minor refers to the determinant of the (n-1)×(n-1) submatrix Mi,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix Mi,j itself). The corresponding cofactor is the signed minor
The adjugate matrix adj M (in older literature called adjoint matrix) is the n×n matrix whose (i,j) entry is the (j,i) cofactor (note the transposition of the indices). Letting In be the n×n identity (unit) matrix, we have
which encodes the rule for expansion of the determinant of M by any the cofactors of any row or column. This expression shows that if det(M) is non-zero, then M is invertible and its inverse is the following,
A proof of this equation may be found in this article.
Consider the following example matrix,
Its minors are the determinants (vertical bars indicate a determinant):
The adjugate matrix of M is
and the inverse matrix is
and the other matrix elements of the product follow likewise.
- ↑ The term "adjoint" for the adjugate matrix is disappearing because it is felt that it is easily confused with Hermitian adjoint, the transpose and complex conjugate of a matrix.
- C.W. Norman (1986). Undergraduate Algebra: A first course. Oxford University Press, 306,310,315. ISBN 0-19-853248-2.
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