The Adjoint of a Matrix

Date: 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/