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```
Date: 10/21/98 at 14:22:30
From: Cliff Free

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

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:

and, dividing this by det(Adj(A))^([n-2]/[n-1]), you get:

comparing this to the first equation above, out will pop:

[-24  6  54]
[  6 -3  -3]

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/
```
Associated Topics:
High School Linear Algebra

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