Let's solve this problem using matrices.

Step 1: Define the Variables

Let:

• $x$ be Amila's take-home salary,
• $y$ be Babu's take-home salary,
• $z$ be Christy's take-home salary.

Step 2: Set Up the Equations

Based on the problem, we have the following system of linear equations:

1. The total salary: $x + y + z = 970$

2. Amila's salary plus twice each of the others: $x + 2y + 2z = 1640$

3. Christy's salary plus twice each of the others: $2x + 2y + z = 1600$

Step 3: Write the System of Equations in Matrix Form

We can represent the system of equations in matrix form as $AX = B$, where:

1 & 1 & 1 \\ 1 & 2 & 2 \\ 2 & 2 & 1 \end{pmatrix} \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \quad B = \begin{pmatrix} 970 \\ 1640 \\ 1600 \end{pmatrix}$$ ### Step 4: Solve for $$X$$ To find $$X$$, we need to calculate $$X = A^{-1}B$$, where $$A^{-1}$$ is the inverse of matrix $$A$$. #### 4.1 Calculate the Determinant of $$A$$ The determinant of $$A$$ is calculated as: $$\text{det}(A) = 1(2 \times 1 - 2 \times 2) - 1(1 \times 1 - 2 \times 2) + 1(1 \times 2 - 2 \times 1)$$ $$\text{det}(A) = 1(-2) - 1(-3) + 1(0) = -2 + 3 + 0 = 1$$ Since the determinant is non-zero, $$A$$ is invertible. #### 4.2 Find the Inverse of $$A$$ The inverse of $$A$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$$ The adjugate of $$A$$ is found by calculating the cofactor matrix and then transposing it. Since $$\text{det}(A) = 1$$, the inverse is simply the adjugate. #### 4.3 Multiply $$A^{-1}$$ by $$B$$ Finally, we multiply the inverse matrix $$A^{-1}$$ by matrix $$B$$ to find the values of $$x$$, $$y$$, and $$z$$. Let me calculate the final result.The take-home salaries of Amila, Babu, and Christy are as follows: - Amila's salary ($$x$$) = Rs. 300 per week - Babu's salary ($$y$$) = Rs. 330 per week - Christy's salary ($$z$$) = Rs. 340 per week Would you like more details or have any questions about the solution? Here are 5 related questions to expand your understanding: 1. How would the solution change if Amila’s salary was different? 2. What if we wanted to solve this problem using substitution or elimination instead of matrices? 3. How can we verify that these salaries satisfy all the original equations? 4. What is the significance of the determinant in finding the inverse of a matrix? 5. Can you apply this method to a 4-variable system of equations? **Tip:** When solving systems of equations, matrices offer a powerful and systematic approach, especially for larger systems with more variables.