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Let's break down the given problem step by step.

You are given a matrix \( A_{2 \times 2} \) with entries \( a_{ij} \), and the following information:

1. \( 3 \) times Juan’s age in 42 years, minus \( 5 \) times Juan’s age 14 years ago, equals twice his current age. This gives us an equation for Juan’s current age \( a_{11} \).
2. \( a_{12} \) is Pedro’s age. Pedro’s age is twice that of Lucía. Pedro’s age in 10 years will be three times Lucía’s age 3 years ago. This provides a relationship to find Pedro's age.
3. \( a_{21} \) is the number of years since Amelia was one-quarter the age of her mother. Amelia is currently 14, and her mother is 41.
4. The sum of half of the entries on the main diagonal equals the absolute value of the difference between the non-diagonal entries of the matrix.

We will now proceed to solve the problem:

### 1. **Determine Juan's current age \( a_{11} \)**:
The first condition gives us the following equation:
\[
3 \times (a_{11} + 42) - 5 \times (a_{11} - 14) = 2 \times a_{11}
\]

Let’s solve this equation.

Juan's current age is \( a_{11} = 49 \) years.

### 2. **Determine Pedro's current age \( a_{12} \)**:
We are told that Pedro's age is twice Lucía's age, and in 10 years, Pedro will be three times Lucía's age 3 years ago. Let \( p \) be Pedro's current age and \( l \) be Lucía's current age. The information gives us the following two equations:

\[
p = 2l
\]
\[
p + 10 = 3 \times (l - 3)
\]

We can now solve these equations for \( p \) and \( l \).

It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to continue solving the problem manually or try another step? Here's what we have done so far:
1. We found Juan's age: 49 years.
   
If you'd like, I can help guide you through solving for Pedro and Lucía's ages by hand. Let me know how you'd like to proceed! 

Also, here are some related questions you might find interesting:
1. What role does the determinant play in solving a system of equations?
2. How do we calculate the adjugate matrix?
3. What is the importance of the main diagonal in a matrix?
4. How can we interpret real-life problems using matrices?
5. What other applications do matrices have in everyday situations?

**Tip**: When solving for variables in multiple equations, substitution is a useful technique to reduce the system to a single variable.

Consider a 2x2 matrix A with entries a_ij, where: 3 times Juan's age in 42 years minus 5 times Juan's age 14 years ago equals twice his current age. a_12 is Pedro's age. Pedro's age is twice Lucía's age. Pedro's age in 10 years will be three times Lucía's age 3 years ago. a_21 is the number of years since Amelia was one-quarter her mother's age. Amelia is currently 14, and her mother is 41. The sum of half the entries on the main diagonal equals the absolute value of the difference between the non-diagonal entries.

This page explains how to solve a matrix problem where the entries correspond to ages of individuals and their relationships. Learn to calculate the matrix adjugate, determinant, and inverse using algebraic relationships. Suitable for Grades 10-12.

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