1. Problem: Find the inverse of matrix $A = \begin{pmatrix}3 & 4 \\ 6 & 7\end{pmatrix}$. Step 1: Calculate the determinant: $\det(A) = 3 \times 7 - 4 \times 6 = 21 - 24 = -3$. Step 2: Inverse formula for 2x2 matrix $A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$ is: $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ Step 3: Applying this for $A$, $$A^{-1} = \frac{1}{-3} \begin{pmatrix}7 & -4 \\ -6 & 3\end{pmatrix} = \begin{pmatrix} -\frac{7}{3} & \frac{4}{3} \\ 2 & -1 \end{pmatrix}$$ --- 3. Problem: Find the inverse of matrix $C = \begin{pmatrix}9 & 2 \\ 3 & 4\end{pmatrix}$. Step 1: Calculate determinant: $\det(C) = 9 \times 4 - 2 \times 3 = 36 - 6 = 30$ Step 2: Calculate inverse: $$C^{-1} = \frac{1}{30} \begin{pmatrix}4 & -2 \\ -3 & 9 \end{pmatrix} = \begin{pmatrix} \frac{2}{15} & -\frac{1}{15} \\ -\frac{1}{10} & \frac{3}{10} \end{pmatrix}$$ --- 5. Problem: Find the inverse of matrix $E = \begin{pmatrix}5 & -3 \\ 4 & -2\end{pmatrix}$. Step 1: Calculate determinant: $\det(E) = 5 \times (-2) - (-3) \times 4 = -10 + 12 = 2$ Step 2: Calculate inverse: $$E^{-1} = \frac{1}{2} \begin{pmatrix} -2 & 3 \\ -4 & 5 \end{pmatrix} = \begin{pmatrix} -1 & \frac{3}{2} \\ -2 & \frac{5}{2} \end{pmatrix}$$ --- 7. Problem: Find the inverse of matrix $G = \begin{pmatrix}-3 & 5 \\ -7 & 6\end{pmatrix}$. Step 1: Calculate determinant: $\det(G) = (-3) \times 6 - 5 \times (-7) = -18 + 35 = 17$ Step 2: Calculate inverse: $$G^{-1} = \frac{1}{17} \begin{pmatrix}6 & -5 \\ 7 & -3 \end{pmatrix} = \begin{pmatrix} \frac{6}{17} & -\frac{5}{17} \\ \frac{7}{17} & -\frac{3}{17} \end{pmatrix}$$ --- 9. Problem: Find the inverse of matrix $I = \begin{pmatrix}-7 & 3 \\ -4 & 2\end{pmatrix}$. Step 1: Calculate determinant: $\det(I) = (-7) \times 2 - 3 \times (-4) = -14 + 12 = -2$ Step 2: Calculate inverse: $$I^{-1} = \frac{1}{-2} \begin{pmatrix}2 & -3 \\ 4 & -7 \end{pmatrix} = \begin{pmatrix} -1 & \frac{3}{2} \\ -2 & \frac{7}{2} \end{pmatrix}$$ --- 10. Problem: Find the inverse of matrix $J = \begin{pmatrix}-7 & -6 \\ -4 & -3\end{pmatrix}$. Step 1: Calculate determinant: $\det(J) = (-7) \times (-3) - (-6) \times (-4) = 21 - 24 = -3$ Step 2: Calculate inverse: $$J^{-1} = \frac{1}{-3} \begin{pmatrix}-3 & 6 \\ 4 & -7 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ -\frac{4}{3} & \frac{7}{3} \end{pmatrix}$$ --- 18. Problem: Find the inverse of matrix $R = \begin{pmatrix}-8 & 3 \\ 5 & -4\end{pmatrix}$. Step 1: Calculate determinant: $\det(R) = (-8) \times (-4) - 3 \times 5 = 32 - 15 = 17$ Step 2: Calculate inverse: $$R^{-1} = \frac{1}{17} \begin{pmatrix}-4 & -3 \\ -5 & -8 \end{pmatrix} = \begin{pmatrix} -\frac{4}{17} & -\frac{3}{17} \\ -\frac{5}{17} & -\frac{8}{17} \end{pmatrix}$$ --- 19. Problem: Find the inverse of matrix $S = \begin{pmatrix}-8 & 4 \\ -3 & 2\end{pmatrix}$. Step 1: Calculate determinant: $\det(S) = (-8) \times 2 - 4 \times (-3) = -16 + 12 = -4$ Step 2: Calculate inverse: $$S^{-1} = \frac{1}{-4} \begin{pmatrix}2 & -4 \\ 3 & -8 \end{pmatrix} = \begin{pmatrix} -\frac{1}{2} & 1 \\ -\frac{3}{4} & 2 \end{pmatrix}$$ --- 20. Problem: Find the inverse of matrix $T = \begin{pmatrix}4 & -6 \\ 3 & -7\end{pmatrix}$. Step 1: Calculate determinant: $\det(T) = 4 \times (-7) - (-6) \times 3 = -28 + 18 = -10$ Step 2: Calculate inverse: $$T^{-1} = \frac{1}{-10} \begin{pmatrix}-7 & 6 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} \frac{7}{10} & -\frac{3}{5} \\ \frac{3}{10} & -\frac{2}{5} \end{pmatrix}$$