Transforming Matrices With Elementary Row Operations A Detailed Analysis

Elementary row operations are fundamental tools in linear algebra, particularly when dealing with matrices. They allow us to manipulate matrices to solve systems of linear equations, find inverses, and determine the rank of a matrix. A common question that arises is whether a given matrix can be transformed into another using a sequence of these operations. In this article, we will delve into this question, specifically addressing whether the matrix

[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]

can be transformed into the matrix

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 0 ]

using elementary row operations. We will explore the underlying principles, demonstrate the operations, and discuss the implications of the result.

Understanding Elementary Row Operations

Before we tackle the specific problem, it's crucial to understand what elementary row operations are and how they affect a matrix. There are three types of elementary row operations:

1. Row Swapping: Interchanging two rows of the matrix.
2. Row Scaling: Multiplying a row by a non-zero scalar.
3. Row Addition: Adding a multiple of one row to another row.

These operations are essential because they do not change the solution set of the system of linear equations represented by the matrix. In other words, applying elementary row operations transforms a matrix into an equivalent matrix. This equivalence is crucial for solving systems of equations and simplifying matrices.

The Role of Elementary Row Operations

Elementary row operations play a pivotal role in transforming matrices into simpler forms, such as row-echelon form or reduced row-echelon form. These forms make it easier to analyze the properties of the matrix, such as its rank and nullity. The rank of a matrix, which is the number of non-zero rows in its row-echelon form, is a key indicator of the matrix's invertibility and the solvability of the corresponding system of linear equations.

Transforming Matrices: A Step-by-Step Approach

The process of transforming a matrix using elementary row operations involves systematically applying these operations to achieve a desired form. This often involves creating leading ones (1s) in each row and zeros below and above these leading ones. The goal is to simplify the matrix while preserving its essential properties. For instance, when solving a system of linear equations, the aim is to transform the augmented matrix into reduced row-echelon form, which directly reveals the solutions.

Analyzing the Given Matrices

Let's consider the given matrices:

Matrix A:

[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]

Matrix B:

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 0 ]

The question is whether we can transform Matrix A into Matrix B using elementary row operations. To answer this, we need to examine the properties of Matrix A and how elementary row operations can affect these properties.

Determinant and Rank

A crucial property to consider is the determinant of Matrix A. The determinant of a 3x3 matrix

[ a b c ]
[ d e f ]
[ g h i ]

is given by a(ei − fh) − b(di − fg) + c(dh − eg). For Matrix A, the determinant is:

1(5*9 − 6*8) − 2(4*9 − 6*7) + 3(4*8 − 5*7) = 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0

Since the determinant of Matrix A is 0, it is a singular matrix, meaning it is not invertible. This implies that the rank of Matrix A is less than 3. The rank of a matrix is the number of linearly independent rows (or columns). In this case, the rows of Matrix A are linearly dependent.

Linear Dependence

The rows of Matrix A are linearly dependent because the third row can be obtained by adding twice the second row to the first row. Specifically:

Row 3 = 2 * Row 2 - Row 1
[ 7 8 9 ] = 2 * [ 4 5 6 ] - [ 1 2 3 ]
[ 7 8 9 ] = [ 8 10 12 ] - [ 1 2 3 ]
[ 7 8 9 ] = [ 7 8 9 ]

This linear dependence is a key factor in determining whether Matrix A can be transformed into Matrix B.

Properties of Matrix B

Matrix B, on the other hand, has a specific structure. It has two leading ones (1s) in the first two rows and a row of zeros. This form suggests that Matrix B has a rank of 2, as there are two linearly independent rows. The determinant of the submatrix formed by the first two rows and columns is:

| 1 0 |
| 0 1 | = 1*1 - 0*0 = 1

which is non-zero, confirming that the first two rows are linearly independent.

Elementary Row Operations and Rank

Elementary row operations do not change the rank of a matrix. This is a fundamental principle in linear algebra. If Matrix A has a rank of r, any matrix obtained from A by elementary row operations will also have a rank of r. Therefore, if we can transform Matrix A into Matrix B, the ranks of both matrices must be the same.

Attempting the Transformation

Now, let's attempt to transform Matrix A into Matrix B using elementary row operations. We'll start by trying to create zeros in the first column below the first element (1):

1. Subtract 4 times the first row from the second row (R2 -> R2 - 4R1):

   [ 1 2 3 ]
   [ 0 -3 -6 ]
   [ 7 8 9 ]
   

2. Subtract 7 times the first row from the third row (R3 -> R3 - 7R1):

   [ 1 2 3 ]
   [ 0 -3 -6 ]
   [ 0 -6 -12 ]
   

Next, we'll try to create a leading one in the second row:

3. Divide the second row by -3 (R2 -> R2 / -3):

   [ 1 2 3 ]
   [ 0 1 2 ]
   [ 0 -6 -12 ]
   

Now, we'll create a zero below the leading one in the second row:

4. Add 6 times the second row to the third row (R3 -> R3 + 6R2):

   [ 1 2 3 ]
   [ 0 1 2 ]
   [ 0 0 0 ]
   

Finally, we'll create a zero above the leading one in the second row:

5. Subtract 2 times the second row from the first row (R1 -> R1 - 2R2):

   [ 1 0 -1 ]
   [ 0 1 2 ]
   [ 0 0 0 ]
   

We have transformed Matrix A into the form:

[ 1 0 -1 ]
[ 0 1 2 ]
[ 0 0 0 ]

This matrix is in reduced row-echelon form. Notice that it has two non-zero rows, which confirms that the rank of the transformed matrix is 2. However, this matrix is not equal to Matrix B.

Conclusion: Can Matrix A Be Transformed into Matrix B?

Based on our analysis and the attempted transformation, we can conclude that Matrix A cannot be transformed into Matrix B using elementary row operations. The key reason is that while both matrices have a rank less than 3, Matrix A has a determinant of 0 and after applying elementary row operations, it can be reduced to a matrix with a rank of 2. Matrix B also has a rank of 2, but the specific arrangement of elements (1s and 0s) in Matrix B is not achievable through elementary row operations from the reduced form of Matrix A.

Final Thoughts

In summary, understanding the properties of matrices, such as their determinant and rank, is crucial in determining whether one matrix can be transformed into another using elementary row operations. While elementary row operations are powerful tools for simplifying matrices and solving systems of equations, they cannot alter the fundamental properties of a matrix, such as its rank. Therefore, in this case, the transformation is not possible.

This exploration underscores the importance of linear algebra concepts in understanding matrix transformations and their implications in various mathematical and computational contexts. The ability to analyze matrices and apply elementary row operations effectively is a valuable skill in many fields, including engineering, computer science, and economics.