1. **State the problem:** Kimi and Jarjit deposited money in two banks, Bank ABC and Bank PQR. We know the amounts deposited and the total interest each received after one year. We need to find the interest rates of both banks using matrix methods. 2. **Define variables:** Let $x$ be the interest rate of Bank ABC and $y$ be the interest rate of Bank PQR. 3. **Set up the system of equations:** From the problem: - Kimi's interest: $2500x + 1500y = 175$ - Jarjit's interest: $3000x + 2000y = 220$ 4. **Write in matrix form:** $$\begin{bmatrix}2500 & 1500 \\ 3000 & 2000\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}175 \\ 220\end{bmatrix}$$ 5. **Calculate the determinant of the coefficient matrix:** $$D = 2500 \times 2000 - 1500 \times 3000 = 5,000,000 - 4,500,000 = 500,000$$ 6. **Find determinants for $x$ and $y$:** $$D_x = \begin{vmatrix}175 & 1500 \\ 220 & 2000\end{vmatrix} = 175 \times 2000 - 1500 \times 220 = 350,000 - 330,000 = 20,000$$ $$D_y = \begin{vmatrix}2500 & 175 \\ 3000 & 220\end{vmatrix} = 2500 \times 220 - 175 \times 3000 = 550,000 - 525,000 = 25,000$$ 7. **Calculate the interest rates:** $$x = \frac{D_x}{D} = \frac{20,000}{500,000} = \frac{\cancel{20,000}}{\cancel{500,000}} = 0.04 = 4\%$$ $$y = \frac{D_y}{D} = \frac{25,000}{500,000} = \frac{\cancel{25,000}}{\cancel{500,000}} = 0.05 = 5\%$$ 8. **Interpretation:** The interest rate for Bank ABC is 4% and for Bank PQR is 5%.