Row Reduction: Adding Rows in Matrix

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In summary: However, you should stick to a specific algorithm, such as dividing the second row by 2 and adding it to the third row, or swapping the second and third rows and adding twice the new second row to the new third row. This ensures consistency and avoids mistakes.
  • #1
AerospaceEng
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quick question if i have the matrix; (R represents a specific row)

1 1 3 -3 0
0 2 1 -3 3
0 -1 -1 2 -1

R3+R2 -->R2

1 1 3 -3 0
0 1 0 -1 2
0 -1 -1 2 -1

and then R2+R3 -->R3

1 1 3 -3 0
0 1 0 -1 2
0 0 -1 1 1

is there anything wrong with my adding of rows or what not? I'm really just second guessing if i can add a lower row to a higher row. thanks.
 
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  • #2
I think it is far better to use a simple "algorithm" always doing things the same way, working on one column at a time, from left to rightm top to bottom.

That said, I would not Use your first step where you add the third row to the second. Instead I would either "divide the second row by 2" then "add the new second row to the third" which gives
[tex]\begin{bmatrix}1 & 1 & 3 & -3 & 0 \\
0 & 1 & 1/2 & -3/2 & 3/2 \\
0 & 0 & -1/2 & 1/2 & 1/2 \end{bmatrix}[/tex]

or "swap the second and third rows" then "add twice the new second row to the new third row".

[tex]\begin{bmatrix}1 & 1 & 3 & -3 & 0 \\
0 & -1 & -1 & 2 & -1 \\
0 & 0 & -1 & 1 & 1\end{bmatrix}[/tex]
 
  • #3
AerospaceEng said:
quick question if i have the matrix; (R represents a specific row)

1 1 3 -3 0
0 2 1 -3 3
0 -1 -1 2 -1

R3+R2 -->R2

1 1 3 -3 0
0 1 0 -1 2
0 -1 -1 2 -1

and then R2+R3 -->R3

1 1 3 -3 0
0 1 0 -1 2
0 0 -1 1 1

is there anything wrong with my adding of rows or what not? I'm really just second guessing if i can add a lower row to a higher row. thanks.

You do add multiple of lower rows to higher rows, after reducing, to turn your matrix
into reduced row-echelon.
 

1. What is row reduction?

Row reduction, also known as Gaussian elimination, is a method used in linear algebra to simplify a system of linear equations. It involves manipulating the rows of a matrix to reduce it to a simpler form, making it easier to solve.

2. Why is row reduction important?

Row reduction is important because it allows us to solve systems of linear equations and find the solutions to real-world problems. It also helps us understand the properties and relationships of matrices, which are essential in various fields such as engineering, physics, and computer science.

3. How do you perform row reduction?

To perform row reduction, we use a set of elementary row operations, which include swapping rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another. By using these operations, we can manipulate the matrix to a simpler form, such as a row-echelon form or reduced row-echelon form.

4. Can row reduction be used for any size matrix?

Yes, row reduction can be used for any size matrix. However, the complexity of the process increases as the size of the matrix increases. For larger matrices, it is often more efficient to use computer software to perform row reduction.

5. What are the applications of row reduction?

Row reduction has various applications in fields such as engineering, physics, and computer science. It is used to solve systems of linear equations, compute determinants and inverses of matrices, and find eigenvalues and eigenvectors. It is also used in machine learning algorithms and image processing techniques.

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