Triangularizing the Given Matrix

Oct 13, 2025 by miodeya.com 43 views

Introduction

In linear algebra, a matrix is said to be triangular if all the elements below or above the main diagonal are zero. Triangularizing a matrix is an essential operation in solving systems of linear equations and finding the inverse of a matrix. In this article, we will explore the process of triangularizing a matrix and determine which row operation will achieve this goal.

What is Triangularization?

Triangularization is the process of transforming a matrix into a triangular form using row operations. There are two types of triangular matrices: upper triangular and lower triangular. An upper triangular matrix has all the elements below the main diagonal as zero, while a lower triangular matrix has all the elements above the main diagonal as zero.

Row Operations

Row operations are the basic building blocks of triangularization. There are three types of row operations:

Row Swap: Swapping two rows of a matrix.
Row Scaling: Multiplying a row of a matrix by a non-zero scalar.
Row Addition: Adding a multiple of one row to another row.

Triangularizing the Given Matrix

The given matrix is:

{ \left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 6 \\ 2 & 0 & 1 & 10 \end{array}\right] \}

To triangularize this matrix, we need to eliminate the elements below the main diagonal. Let's analyze the options:

Option A: R1 - R3

Performing the row operation R1 - R3, we get:

{ \left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 6 \\ 0 & 0 & -1 & 8 \end{array}\right] \}

This operation eliminates the element below the main diagonal in the first column. However, it introduces a non-zero element above the main diagonal in the third column.

Option B: 2R2 - R3

Performing the row operation 2R2 - R3, we get:

{ \left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 6 \\ 2 & 0 & 1 & 10 \end{array}\right] \}

This operation does not eliminate any elements below the main diagonal.

Option C: -2R1 + R3

Performing the row operation -2R1 + R3, we get:

{ \left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 6 \\ 0 & 0 & -1 & 8 \end{array}\right] \}

This operation eliminates the element below the main diagonal in the first column.

Conclusion

Based on the analysis, we can conclude that the correct row operation to triangularize the given matrix is:

Option A: R1 - R3

This operation eliminates the element below the main diagonal in the first column and introduces a non-zero element above the main diagonal in the third column. However, it is the first step towards triangularizing the matrix.

Final Answer

Introduction

In our previous article, we explored the process of triangularizing a matrix and determined which row operation will achieve this goal. In this article, we will answer some frequently asked questions about triangularizing a matrix.

Q: What is the purpose of triangularizing a matrix?

A: Triangularizing a matrix is an essential operation in solving systems of linear equations and finding the inverse of a matrix. It helps to simplify the matrix and make it easier to work with.

Q: What are the types of triangular matrices?

A: There are two types of triangular matrices: upper triangular and lower triangular. An upper triangular matrix has all the elements below the main diagonal as zero, while a lower triangular matrix has all the elements above the main diagonal as zero.

Q: What are the three types of row operations?

A: The three types of row operations are:

Row Swap: Swapping two rows of a matrix.
Row Scaling: Multiplying a row of a matrix by a non-zero scalar.
Row Addition: Adding a multiple of one row to another row.

Q: How do I triangularize a matrix using row operations?

A: To triangularize a matrix using row operations, you need to eliminate the elements below the main diagonal. You can do this by performing row operations such as row swap, row scaling, and row addition.

Q: What is the difference between row swap and row addition?

A: Row swap involves swapping two rows of a matrix, while row addition involves adding a multiple of one row to another row.

Q: Can I triangularize a matrix using only row addition?

A: No, you cannot triangularize a matrix using only row addition. You need to use a combination of row swap, row scaling, and row addition to eliminate the elements below the main diagonal.

Q: How do I know which row operation to use?

A: To determine which row operation to use, you need to analyze the matrix and identify the elements that need to be eliminated. You can use the following steps:

Identify the elements below the main diagonal that need to be eliminated.
Determine which row operation will eliminate these elements.
Perform the row operation to eliminate the elements.

Q: Can I triangularize a matrix with a non-zero element on the main diagonal?

A: No, you cannot triangularize a matrix with a non-zero element on the main diagonal. The main diagonal element must be zero for the matrix to be triangular.

Q: How do I know if a matrix is triangular?

A: To determine if a matrix is triangular, you need to check if all the elements below or above the main diagonal are zero. If they are, then the matrix is triangular.

Conclusion

Triangularizing a matrix is an essential operation in linear algebra. By understanding the types of triangular matrices, row operations, and how to triangularize a matrix, you can simplify complex matrices and make them easier to work with. We hope this Q&A guide has helped you understand the process of triangularizing a matrix.

Final Answer

The final answer is: Triangularizing a matrix is an essential operation in linear algebra that involves eliminating elements below the main diagonal using row operations.