1. **State the problem:** We are given two matrices: $$A = \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}$$ and $$B = \begin{bmatrix} 0.5 & 0.5 \\ 0 & -1 \end{bmatrix}$$ We need to determine if these matrices are inverses of each other. 2. **Recall the definition of inverse matrices:** Two matrices $A$ and $B$ are inverses if and only if $$AB = BA = I$$ where $I$ is the identity matrix. 3. **Calculate the product $AB$:** $$AB = \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0.5 & 0.5 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 2 \times 0.5 + 1 \times 0 & 2 \times 0.5 + 1 \times (-1) \\ 0 \times 0.5 + (-1) \times 0 & 0 \times 0.5 + (-1) \times (-1) \end{bmatrix} = \begin{bmatrix} 1 & 1 - 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ 4. **Calculate the product $BA$:** $$BA = \begin{bmatrix} 0.5 & 0.5 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0.5 \times 2 + 0.5 \times 0 & 0.5 \times 1 + 0.5 \times (-1) \\ 0 \times 2 + (-1) \times 0 & 0 \times 1 + (-1) \times (-1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ 5. **Conclusion:** Since both $AB$ and $BA$ equal the identity matrix $I$, matrices $A$ and $B$ are inverses of each other. **Final answer:** Yes, these matrices are inverses.