Linear Alg: Find all values of a and determine all solutions

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In summary, the conversation discusses a system of linear equations and the reduced form of the matrix representing the equations. The goal is to find the solutions for x, y, and z. The expert suggests looking at the relationships shown in rows 1 and 3 of the reduced matrix.
  • #1
sumtingwong59
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Homework Statement



x - 2y + z = 0
x + ay - 3z = 0
-x + 6y - 5z = 0

Homework Equations


3. The Attempt at a Solution [/B]

i've got it reduced all the way down to
1 0 0 0
0 1 -1 0
0 0 -1a+3 0

I could show more of the steps if needed, but I am just wondering what I can do to reduce the matrix further, or how i could get solutions from what i have so far
 
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  • #2
I think your reduced form is off.

I get

1 0 -1
0 a-2 0
0 1 -1If you want to find the solutions then you need to figure out what this matrix is telling you. Start with rows 1 and 3. What do they tell you about the relationship between x, y and z? If you can figure out those then we can move onto the second row which is slightly different.
 

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is important because it provides tools for solving complex problems in various fields such as engineering, physics, economics, and computer science.

2. What are the basic concepts in linear algebra?

The basic concepts in linear algebra include vectors, matrices, linear transformations, and systems of linear equations. These concepts are used to represent and solve problems involving linear relationships.

3. How do you find all values of a in a linear algebra problem?

To find all values of a in a linear algebra problem, you can use the technique of substitution or elimination. This involves manipulating the given equations to isolate the variable a and solve for its value.

4. What are the different types of solutions in linear algebra?

The different types of solutions in linear algebra are unique, no solution, and infinite solutions. A unique solution exists when there is a single set of values that satisfies all the given equations. No solution exists when the equations are inconsistent and contradict each other. Infinite solutions exist when the equations are dependent and represent the same line or plane.

5. How do you determine all solutions in a linear algebra problem?

To determine all solutions in a linear algebra problem, you can use the technique of Gaussian elimination or matrix inversion. These methods involve transforming the given equations into an augmented matrix and performing row operations to obtain the solutions.

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