# Inverse Matrix Question

1. Dec 1, 2008

### lubricarret

1. The problem statement, all variables and given/known data

Let A be an invertible 3x3 matrix. Suppose it is known that:
A =
[u v w
3 3 -2
x y z]
[a 3 b
-1 1 2
c -2 d]
Find det(A)

2. Relevant equations

3. The attempt at a solution

I found A^(-1) to be equal to

So, then re-arranging the formula;
So then det(A) =
[u v w
3 3 -2
x y z]
*
[a 3 b
-1 1 2
c -2 d]

I know the solution to this problem is det(A) = 16. Therefore it must be that
[3 3 -2] * [3 1 2] = det(A)

But, what I am confused about is:
Why is the det equal only to the position (2,2) of the matrix A*adj(A)? As the solution is taken as the position a_(2,2)...

Thanks!

2. Dec 1, 2008

### Dick

If I*det(A)=A*adj(A) then you can evaluate any diagonal element to figure out det(A). a_(2,2) just happens to be the one you can figure out that doesn't have any unknowns in it. No big mysteries here.

3. Dec 1, 2008

### lubricarret

Thanks again Dick for clearing this up for me!