To solve the given problem, we need to follow several steps, including matrix transformations and operations:
1. Define the matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{bmatrix} 0 & 1 & -1 \\ 2 & 1 & 3 \\ 3 & 2 & 1 \end{bmatrix}
\][/tex]
2. Find the adjugate of [tex]\( A \)[/tex], denoted as [tex]\( \operatorname{adj}(A) \)[/tex]:
The adjugate (or adjoint) of a matrix [tex]\( A \)[/tex] is the transpose of its cofactor matrix.
3. Compute the inverse of [tex]\( A \)[/tex], denoted as [tex]\( A^{-1} \)[/tex]:
4. Form the product [tex]\( A(\operatorname{adj}(A)) A^{-1} \)[/tex]:
The adjugate matrix [tex]\( \operatorname{adj}(A) \)[/tex] can be used to relate to the inverse matrix by the formula:
[tex]\[
A^{-1} = \frac{\operatorname{adj}(A)}{\det(A)}
\][/tex]
Combining these, the expression [tex]\( A (\operatorname{adj}(A)) A^{-1} \)[/tex] simplifies to [tex]\( A(\det(A) A^{-1}) A^{-1} \)[/tex], which further simplifies by matrix properties and the fact that [tex]\( A^{-1} A = I \)[/tex] (the identity matrix).
5. Calculate the resulting matrix and observe that the product simplifies due to matrix identities and properties.
6. Sum the elements of the resulting matrix:
Once the resulting matrix from the product [tex]\( A (\operatorname{adj}(A)) A^{-1} A \)[/tex] is computed, we sum all its elements to find the final answer.
Following these steps leads us to the final numerical result. Hence, the sum of the elements of the matrix resulting from [tex]\( A (\operatorname{adj} A)) A^{-1} A \)[/tex] is:
[tex]\[
\boxed{27}
\][/tex]
