What are the steps to solve a system of equations using matrices?

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In summary, the conversation discusses the finding of nonzero scalars a, b, and c that satisfy the equation aA + b(A-B) + c(A+B) = 0 for every pair of vectors A and B. The participants mention the possibility of a general solution for all vectors A and B, but conclude that it is impossible. They then explore the possibility of finding specific values for a, b, and c that would make the equation true for given vectors A and B. One possible answer is a=-2, b=c=1. The conversation also touches on the importance of clarifying the exact statement of the problem in order to find the correct solution.
  • #1
Iyafrady
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Need some help to solve this problem.I have tried using a system of equation(matrices) but hasnt worked out.

Find nonzero scalars a,b and c such that aA+b(A-B)+c(A+B)=0 for every pair of vectors A and B.

Thanks for the help.
 
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  • #2
What do you mean by "vectors"? If A and B are vectors, and they are orthogonal to each other, I don't believe there is a general solution.

EDIT -- oops, yeah there still is a set of solutions. Just distribute the terms, and gather the terms multiplying A and those multiplying B. Assume that A and B are indeed orthogonal. What can you say about those two sets of terms...?
 
  • #3
For every pair of vectors A and B? Do you mean one set of numbers a, b, c such that that is true for all vectors? Obviously that is impossible.

If it were true for all A, B, it must be true for any A and B= 0. Then you must have (a+ b+ c)A= 0 for any A, that is a+ b+ c= 0.

On the other hand, if A= B, you have (a+ 2c)A= 0 so a+ 2c= 0. If A= -B, you have (a+ 2b)A= 0 so a+ 2b= 0. The only numbers that satisfy those three equations are a= b= c= 0.

If, however, you mean find a, b, c so that aA+ b(A- B)+ c(A+ B)= 0 for specific A, B, then you need to have (a+ b+ c)A+ (a- b+ c)B. There will be an infinite number of values of a, b, c such that that is true for any given A and B.
 
  • #4
Hmm, the book says one possible answer is a=-2,b=c=1
 
  • #5
What is the exact statement of the problem from your book? Does it says anything about the vectors A,B, independent or something else?
 
  • #6
what i posted is the EXACT statement, its from a vector analysis course.
 
  • #7
Ok, then! If A,B are arbitrary vectors I would write

<< exact solution edited out by berkeman, but too late to keep the OP from seeing it >>

yielding the book's solution, but I assumed that A,B are arbitrary vectors.
 
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  • #8
I hope nobody succeeded seeing my previous post, which I deleted quickly. I cannot believe how frozen my brains were...
 
  • #9
Ahh!cant believe this easy question gave me problems..so that answer is one of many possible answers just like the book says!..thnx for the help!
 
  • #10
jostpuur said:
I hope nobody succeeded seeing my previous post, which I deleted quickly. I cannot believe how frozen my brains were...

Only the Mentors can see it. We're wispering about it now... :bugeye:
 

FAQ: What are the steps to solve a system of equations using matrices?

What are systems of equations?

Systems of equations refer to a set of two or more equations that are solved simultaneously to find the values of the variables that satisfy all of the equations.

Why do we use systems of equations?

Systems of equations are used to model and solve real-world problems that involve multiple variables and relationships between them. They allow us to find the values of the variables that satisfy all of the given conditions.

What are the different methods for solving systems of equations?

The most commonly used methods for solving systems of equations are substitution, elimination, and graphing. Other methods include Gaussian elimination, Cramer's rule, and matrices.

How do I know which method to use when solving a system of equations?

The method you choose depends on the form of the equations and personal preference. For example, substitution is useful when one of the equations can be easily solved for a variable, while elimination is more efficient when the coefficients of one variable are the same in both equations.

Can systems of equations have more than two variables?

Yes, systems of equations can have any number of variables. However, they become more complex to solve as the number of variables increases, and often require more advanced methods such as matrices or technology.

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