Linear Algebra Row Reduction: Solving a System of Equations with Row Operations

In summary, the conversation is about solving a system of equations using row reduction in order to put it into echelon form. The person has attempted to solve the system, but is struggling to remove a specific number without making another one zero. They also mention that the problem may have multiple solutions or no solution at all. They ask for help in determining the case for their specific problem.
  • #1
SMA_01
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Linear Algebra Row Reduction?

Homework Statement



This question is frustrating me, and I might be missing the obvious, I tend to make algebraic mistakes, but I don't know what I'm doing wrong here!

I'm trying to solve this system using row reduction:

x-5y+4z=-3
2x-7y+3z=-2
-2x+y+7z=-1

Note this does not have to be in reduced echelon form, just regular echelon form.





The Attempt at a Solution




Okay so I started like this:
Note: "l" indicated the bar representing "=" sign
1 -5 4 l -3
2 -7 3 l -2 --> I then used -2(2nd row)+1st row
-2 1 7 l -1 --> And 2(1st row)+ 3rd row

To get:

1 -5 4 l -3
0 3 -5 l 4
0 -9 15 l -7
I'm trying to remove the -9 in the third row but can't seem to do it without the 15 becoming zero.

Any help please?
 
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  • #2


HINT: Some problems have more then 1 solution, some don't have any. [what is yours case?]
 

FAQ: Linear Algebra Row Reduction: Solving a System of Equations with Row Operations

1. What is row reduction in linear algebra?

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 transform it into an equivalent matrix with a simpler form, typically a matrix with a lower triangular or diagonal form.

2. Why is row reduction important in linear algebra?

Row reduction is important in linear algebra because it allows us to solve systems of linear equations, find inverse matrices, and determine the rank of a matrix. It also helps us to understand the relationships between the rows of a matrix and to identify linearly independent rows.

3. How do you perform row reduction?

To perform row reduction, you typically start with a matrix of coefficients and a vector on the right-hand side representing the constants. Then, you use a series of elementary row operations (such as swapping rows, multiplying a row by a constant, or adding a multiple of one row to another) to transform the matrix into its reduced form. The goal is to create an upper triangular matrix with zeros in the lower left corner.

4. What is the purpose of row reduction in solving linear systems?

The purpose of row reduction in solving linear systems is to simplify the system of equations and make it easier to determine the solution. By reducing the matrix to its reduced form, we can easily identify the values of variables that satisfy the equations.

5. Can row reduction be used for any size of matrix?

Yes, row reduction can be used for any size of matrix. However, as the size of the matrix increases, the computation involved in row reduction becomes more complex and time-consuming. In some cases, it may also be difficult to obtain the reduced form of the matrix due to the large number of operations involved.

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