Calculate The Sum Of The Main Diagonal Elements Of A 3x3 Matrix

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A matrix A of dimension 3x3 has the formation law aij = 5 * i * j². What is the sum of the elements of the main diagonal of this matrix? Show how to calculate the elements.

In this article, we will delve into a specific mathematical problem involving matrices. We will explore how to calculate the sum of the elements present on the main diagonal of a 3x3 matrix, given a specific rule for its formation. This type of problem is fundamental in linear algebra and often appears in various mathematical and computational contexts. Understanding how to solve it requires a grasp of matrix structure, element indexing, and basic arithmetic. Let's dissect the problem and arrive at the correct solution, providing a step-by-step explanation for clarity.

Understanding the Problem

The problem presents us with a 3x3 matrix, which we will call A. The defining characteristic of this matrix is its 'law of formation,' given as aij = 5ij². This formula dictates how each element (aij) within the matrix is calculated. The subscripts 'i' and 'j' represent the row and column number of the element, respectively. So, a12 refers to the element in the first row and second column. The task is to find the sum of the elements that lie on the main diagonal of this matrix. The main diagonal consists of elements where the row number equals the column number (i.e., a11, a22, and a33). We are provided with multiple-choice options, suggesting a single numerical answer.

Breaking Down the Law of Formation

The core of solving this problem lies in understanding and applying the given law of formation: aij = 5ij². This formula tells us exactly how to calculate the value of any element in the matrix. Let's break it down further:

  • aij: This represents the element located at the i-th row and the j-th column of the matrix A.
  • 5: This is a constant factor that will be multiplied in the calculation.
  • ij²: This is the crucial part of the formula. It indicates that we need to multiply the row number (i) by the square of the column number (j²). This operation determines the unique value of each element based on its position within the matrix.

To illustrate, let's calculate a few example elements:

  • For a11 (first row, first column), i = 1 and j = 1. Therefore, a11 = 5 * 1 * 1² = 5.
  • For a23 (second row, third column), i = 2 and j = 3. Therefore, a23 = 5 * 2 * 3² = 5 * 2 * 9 = 90.
  • For a32 (third row, second column), i = 3 and j = 2. Therefore, a32 = 5 * 3 * 2² = 5 * 3 * 4 = 60.

By understanding this law, we can systematically calculate any element of the matrix A, which is essential for finding the elements on the main diagonal.

Identifying the Main Diagonal Elements

The main diagonal of a square matrix is a special set of elements. It consists of those elements where the row number and the column number are the same. In a 3x3 matrix, these elements are:

  • a11: The element in the first row and the first column.
  • a22: The element in the second row and the second column.
  • a33: The element in the third row and the third column.

These are the elements we need to calculate using the law of formation (aij = 5ij²) and then sum them up to find the final answer. The other elements of the matrix (a12, a13, a21, a23, a31, a32) are not relevant to this particular problem, although they would be important if we were asked to perform other matrix operations, such as finding the trace (sum of diagonal elements), determinant, or inverse.

Calculating the Diagonal Elements

Now, let's calculate the values of the main diagonal elements using the given law of formation, aij = 5ij²:

  1. Calculate a11:

    • i = 1, j = 1
    • a11 = 5 * 1 * 1² = 5 * 1 * 1 = 5

  2. Calculate a22:

    • i = 2, j = 2
    • a22 = 5 * 2 * 2² = 5 * 2 * 4 = 40

  3. Calculate a33:

    • i = 3, j = 3
    • a33 = 5 * 3 * 3² = 5 * 3 * 9 = 135

Therefore, the elements on the main diagonal of the matrix A are 5, 40, and 135.

Summing the Diagonal Elements

The final step in solving the problem is to add the diagonal elements we just calculated. This will give us the sum of the main diagonal, which is the answer we are looking for.

Sum = a11 + a22 + a33 Sum = 5 + 40 + 135 Sum = 180

Therefore, the sum of the elements on the main diagonal of the matrix A is 180.

Analyzing the Options and Identifying the Correct Answer

Now that we have calculated the sum of the main diagonal elements to be 180, let's revisit the multiple-choice options provided in the problem statement:

A) 30 B) 45 C) 60 D) 75

Our calculated sum of 180 does not match any of the given options. This discrepancy indicates a potential error in the options provided in the original problem statement. The correct answer, based on our calculations using the provided law of formation, is 180.

Addressing a Potential Error in the Problem Statement

It's important to recognize that errors can sometimes occur in problem statements. In this case, the multiple-choice options do not include the correct answer (180), which we derived through a careful step-by-step calculation. When encountering such situations, it's crucial to:

  1. Double-Check Calculations: Ensure that all calculations performed are accurate and that no steps were missed or incorrectly executed. We have thoroughly reviewed our calculations and are confident in the result of 180.

  2. Re-examine the Law of Formation: Verify that the law of formation (aij = 5ij²) was correctly interpreted and applied. We have carefully broken down the formula and applied it to each diagonal element.

  3. Consider Possible Typos: It's possible that there is a typographical error in the provided options. If this were an exam or assignment, it would be prudent to bring this discrepancy to the attention of the instructor or exam proctor.

While we cannot definitively correct the options provided, our analysis clearly demonstrates that the sum of the main diagonal elements, calculated using the given law of formation, is 180.

Conclusion

In this article, we have successfully tackled the problem of finding the sum of the main diagonal elements of a 3x3 matrix, defined by the law of formation aij = 5ij². Through a systematic approach, we:

  1. Understood the problem statement and the significance of the law of formation.
  2. Calculated each element on the main diagonal (a11, a22, a33) using the formula.
  3. Summed these elements to arrive at a final answer of 180.
  4. Identified a discrepancy between our calculated answer and the provided multiple-choice options, highlighting the importance of critical evaluation and error recognition in problem-solving.

This exercise demonstrates the fundamental principles of matrix manipulation and the application of algebraic formulas. While the original problem statement might contain an error in the provided options, the process of solving the problem remains valuable for reinforcing these core mathematical concepts. Understanding matrices and their properties is crucial in various fields, including computer science, engineering, physics, and economics. Matrix operations are used extensively in areas like data analysis, computer graphics, and solving systems of linear equations.