The Adjoint of a MatrixDate: 10/21/98 at 14:22:30 From: Cliff Free Subject: Adjoint of a Matrix Given the adjoint of matrix A, how would you determine matrix A? For example, if: | 4 5 6 | Adj(A) = | 7 8 -3 | | 1 2 3 | then what is A? Date: 10/21/98 at 15:43:04 From: Doctor Rob Subject: Re: Adjoint of a Matrix The equation governing adjoints is: A*Adj(A) = det(A)*I. From this you can determine that det(Adj(A)) = det(A)^(n-1), where n is the number of rows and columns in your matrices. In your example, you know that det(A)^2 = det(Adj(A)) = 36, and n = 3, so det(A) = 6 or -6. In general you will have n-1 possible values of the determinant of A. You can take the adjoint of Adj(A) to get a matrix that is related to A: Adj(A)*Adj(Adj(A)) = det(Adj(A))*I and, dividing this by det(Adj(A))^([n-2]/[n-1]), you get: Adj(A)*[Adj(Adj(A))/det(Adj(A))^([n-2]/[n-1])] = det(A)*I comparing this to the first equation above, out will pop: A = Adj(Adj(A))/det(Adj(A))^([n-2]/[n-1]) In your example, Adj(A) given above, you will find that: Adj(Adj(A)) = [ 30 -3 -63] [-24 6 54] [ 6 -3 -3] det(Adj(A))^(1/2) = 6 or -6 A = +/-[ 5 -1/2 -21/2] [-4 1 9 ] [ 1 -1/2 -1/2] These are the matrices A whose adjoints are the given matrix. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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