This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Math 313 Lecture #11 § 2.3: Cramer’s Rule The Adjoint of a Matrix. In addition to defining and computing the determinant of a matrix, the cofactors of a matrix also play a role in another formula for the inverse of an invertible matrix. The adjoint of an n × n matrix A is the n × n matrix adj( A ) whose ( i, j ) entry is the cofactor A ji of A : adj A = A 11 A 21 . . . A n 1 A 12 A 22 . . . A n 2 . . . . . . . . . . . . A 1 n A 2 n . . . A nn . Notice that adj( A ) is the transpose of the matrix whose ( i, j ) entry is A ij . To see how this is related to the inverse of A when A is invertible, we need to recall the “technical lemma” from last time: a i 1 A j 1 + a i 2 A j 2 + · · · + a in A jn = det( A ) if i = j, if i 6 = j. The ( i, j ) entry of the product A (adj A ) is n X k =1 a ik A jk = det( A ) if i = j, if i 6 = j. Therefore each diagonal entry of A (adj A ) is det( A ) and the entries off the diagonal are zero: A (adj A ) = det(...
View Full Document