For a 3×33 \times 33×3 matrix AAA, if adj(A)=(100010001)\text{adj}(A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}adj(A)=100010001, what is AAA?
III
000
2I2I2I
12I\frac{1}{2}I21I