To find the inverse of the given $2 \times 2$ matrix $M = \begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix}$, we will use the formula for the inverse of a $2 \times 2$ matrix:

$M^{-1} = \frac{1}{\text{det}(M)} \cdot \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

where the matrix $M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, and $\text{det}(M) = ad - bc$.

Step-by-step Calculation:

1. Identify the elements: $a = 0, \, b = -1, \, c = 1, \, d = 2$

2. Compute the determinant: $\text{det}(M) = ad - bc = (0)(2) - (-1)(1) = 0 + 1 = 1$

3. Verify determinant is nonzero: Since $\text{det}(M) = 1 \neq 0$, the matrix is invertible.

4. Apply the inverse formula: Substitute into the formula: $M^{-1} = \frac{1}{1} \cdot \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}$

   Simplify: $M^{-1} = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}$

Final Answer:

The inverse of $M$ is: $M^{-1} = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}$

Let me know if you would like further clarification or additional examples.

Questions to explore:

1. What happens if the determinant of a matrix is zero?
2. How is the formula for the inverse of a $2 \times 2$ matrix derived?
3. How does matrix inversion apply to solving linear systems of equations?
4. Can all square matrices be inverted?
5. How does this method extend to matrices larger than $2 \times 2$?

Tip:

Always check the determinant first to confirm if the matrix is invertible!