Linear Algebra: use elem. row ops to convert A into B

In summary, to show that the given matrices a and b are row equivalent, a sequence of elementary row operations was performed on matrix a to convert it into matrix b. The sequence of operations involved combining rows, swapping rows, and multiplying rows by a scalar. However, the resulting matrix is not equivalent to matrix b, and further operations would be needed to reach the desired solution.
  • #1
leo255
57
2

Homework Statement



Show that the given matrices are row equivalent and find a sequence of elementary row ops that will convert A into B.

a =
2 0 -1
1 1 0
-1 1 1

b =
3 1 -1
3 5 1
2 2 0

Homework Equations

The Attempt at a Solution


[/B]
I apologize in advance, as I'm not sure how to make these matrices look good (will need to look up LATEX, so I can present my matrices better).

The elementary row ops that I chose to do were as follows:

R2 + R1 --> R1
R2 <---> R3
R3 + R2 -->R2
R1 + R2 --> R2
2R3 --> R3
R3 + R2 --> R2

This gave me the following matrix, which is not the answer, but is my best try:

a =
3 1 -1
5 5 0
2 2 1
 
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  • #2
leo255 said:
R2 + R1 --> R1
R2 <---> R3
R3 + R2 -->R2
R1 + R2 --> R2
2R3 --> R3
R3 + R2 --> R2
Close..
R2 + R1 →R1 OK
4*R2 + R3→R2
Now do the third row...
 

Related to Linear Algebra: use elem. row ops to convert A into B

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with systems of linear equations and the properties of linear transformations. It involves the study of vectors, matrices, and their operations, and has various applications in fields such as physics, engineering, and computer science.

2. What are elementary row operations?

Elementary row operations are three basic operations that can be performed on the rows of a matrix: 1) multiplying a row by a non-zero constant, 2) adding a multiple of one row to another row, and 3) swapping two rows. These operations are used to manipulate a matrix and solve systems of linear equations.

3. Why do we use elementary row operations to convert a matrix?

Elementary row operations are useful for converting a matrix into a specific form, such as a reduced row-echelon form, which can make it easier to solve systems of linear equations or perform other operations on the matrix. These operations do not change the solutions to the system of equations, but they can make the calculations more efficient.

4. What is the purpose of converting matrix A into matrix B?

Converting matrix A into matrix B can help us to simplify the matrix and make it easier to perform calculations or solve problems involving the matrix. It can also help us to identify important properties of the matrix, such as the rank, determinant, and inverse, which can have various applications in linear algebra and other fields.

5. Can we use elementary row operations to convert any matrix into any other matrix?

Yes, elementary row operations can be used to convert any matrix into any other matrix. However, the resulting matrix may not always be a unique solution, and it depends on the specific operations performed and the desired form of the matrix. In some cases, it may not be possible to convert a matrix into a desired form using these operations.

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