1. **State the problem:** We want to verify if the matrix $\begin{bmatrix} 2/17 & -5/7 \\ -1/17 & 6/7 \end{bmatrix}$ is the inverse of $\begin{bmatrix} 6 & -5 \\ -1 & -2 \end{bmatrix}$ by checking if their product equals the identity matrix $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. 2. **Recall the formula for matrix multiplication:** For two 2x2 matrices $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}$, their product is $$AB = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$ 3. **Calculate each entry of the product:** - Top-left entry: $$6 \times \frac{2}{17} + (-5) \times \left(-\frac{1}{17}\right) = \frac{12}{17} + \frac{5}{17} = \frac{17}{17} = 1$$ - Top-right entry: $$6 \times \left(-\frac{5}{7}\right) + (-5) \times \frac{6}{7} = -\frac{30}{7} - \frac{30}{7} = -\frac{60}{7} \neq 0$$ - Bottom-left entry: $$-1 \times \frac{2}{17} + (-2) \times \left(-\frac{1}{17}\right) = -\frac{2}{17} + \frac{2}{17} = 0$$ - Bottom-right entry: $$-1 \times \left(-\frac{5}{7}\right) + (-2) \times \frac{6}{7} = \frac{5}{7} - \frac{12}{7} = -\frac{7}{7} = -1 \neq 1$$ 4. **Interpretation:** The product matrix is $$\begin{bmatrix} 1 & -\frac{60}{7} \\ 0 & -1 \end{bmatrix}$$ which is not the identity matrix. 5. **Conclusion:** The given second matrix is not the inverse of the first matrix because the product is not the identity matrix. 6. **Find the correct inverse:** - The inverse of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ - Calculate the determinant: $$\det(A) = 6 \times (-2) - (-5) \times (-1) = -12 - 5 = -17$$ - Compute the inverse: $$A^{-1} = \frac{1}{-17} \begin{bmatrix} -2 & 5 \\ 1 & 6 \end{bmatrix} = \begin{bmatrix} \frac{2}{17} & -\frac{5}{17} \\ -\frac{1}{17} & -\frac{6}{17} \end{bmatrix}$$ 7. **Verify the inverse:** Multiply $A$ and $A^{-1}$ to confirm it equals the identity matrix. **Final answer:** The correct inverse matrix is $$\boxed{\begin{bmatrix} \frac{2}{17} & -\frac{5}{17} \\ -\frac{1}{17} & -\frac{6}{17} \end{bmatrix}}$$