Find the inverse of the following matrix $M$, if possible. \[ M=\left[\begin{array}{rrr} 1 & 2 & 0 \\ -1 & 2 & 1 \\ 0 & 3 & 1 \end{array}\right] \]

Step 1 ：Given the matrix $M$ as

Step 2 ：\[M=\left[\begin{array}{rrr}1 & 2 & 0 \-1 & 2 & 1 \0 & 3 & 1\end{array}\right]\]

Step 3 ：The inverse of a matrix $A$ is a matrix $A^{-1}$ such that when you multiply $A$ and $A^{-1}$, you get the identity matrix. The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.

Step 4 ：To find the inverse of the matrix $M$, we calculate it to be

Step 5 ：\[M^{-1}=\left[\begin{array}{rrr}-1 & -2 & 2 \1 & 1 & -1 \-3 & -3 & 4\end{array}\right]\]

Step 6 ：\(\boxed{M^{-1}=\left[\begin{array}{rrr}-1 & -2 & 2 \1 & 1 & -1 \-3 & -3 & 4\end{array}\right]}\)