1. **Stating the problem:** (a) Given matrix $$H=\begin{pmatrix}2 & 6 \\ 1 & k\end{pmatrix}$$, find the value of $$k$$ such that the inverse of $$H$$ does not exist. 2. **Formula and rule:** The inverse of a 2x2 matrix $$A=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$$ exists if and only if its determinant $$\det(A) = ad - bc \neq 0$$. 3. **Calculate determinant of $$H$$:** $$\det(H) = (2)(k) - (6)(1) = 2k - 6$$ 4. **Condition for no inverse:** Inverse does not exist if $$\det(H) = 0$$. 5. **Solve for $$k$$:** $$2k - 6 = 0$$ $$2k = 6$$ $$\cancel{2}k = \cancel{2}3$$ $$k = 3$$ --- 1. **Stating the problem:** (b)(i) Puan Mariam and her son's ages sum to 34 this year. Three years later, Puan Mariam's age is triple her son's age. Find their current ages using matrix method. 2. **Define variables:** Let $$x$$ = Puan Mariam's current age, $$y$$ = son's current age. 3. **Form equations:** - Sum of ages: $$x + y = 34$$ - After 3 years: $$x + 3 = 3(y + 3)$$ 4. **Rewrite second equation:** $$x + 3 = 3y + 9$$ $$x - 3y = 6$$ 5. **Matrix form:** $$\begin{pmatrix}1 & 1 \\ 1 & -3\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}34 \\ 6\end{pmatrix}$$ 6. **Calculate determinant of coefficient matrix:** $$\det = (1)(-3) - (1)(1) = -3 - 1 = -4$$ 7. **Find inverse matrix:** $$A^{-1} = \frac{1}{-4} \begin{pmatrix}-3 & -1 \\ -1 & 1\end{pmatrix} = \begin{pmatrix}\frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & -\frac{1}{4}\end{pmatrix}$$ 8. **Multiply inverse by constants:** $$\begin{pmatrix}x \\ y\end{pmatrix} = A^{-1} \begin{pmatrix}34 \\ 6\end{pmatrix} = \begin{pmatrix}\frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & -\frac{1}{4}\end{pmatrix} \begin{pmatrix}34 \\ 6\end{pmatrix}$$ 9. **Calculate:** $$x = \frac{3}{4} \times 34 + \frac{1}{4} \times 6 = \frac{102}{4} + \frac{6}{4} = \frac{108}{4} = 27$$ $$y = \frac{1}{4} \times 34 - \frac{1}{4} \times 6 = \frac{34}{4} - \frac{6}{4} = \frac{28}{4} = 7$$ 10. **Answer:** Puan Mariam is 27 years old and her son is 7 years old this year. --- 1. **Stating the problem:** (b)(ii) Puan Mariam plans to retire at 55 years old; her son will finish studies at 25 years old. Based on (b)(i), will her son finish studies before she retires? 2. **From (b)(i):** Puan Mariam's current age = 27 Son's current age = 7 3. **Calculate years until retirement and finishing studies:** - Years until Puan Mariam retires: $$55 - 27 = 28$$ years - Years until son finishes studies: $$25 - 7 = 18$$ years 4. **Compare:** Son finishes studies in 18 years, Puan Mariam retires in 28 years. 5. **Conclusion:** Since 18 < 28, the son will finish his studies before Puan Mariam retires. **Final answers:** (a) $$k = 3$$ (b)(i) Puan Mariam's age = 27, Son's age = 7 (b)(ii) Yes, the son will finish studies before Puan Mariam retires.