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

[Iyafrady]

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.

 

[berkeman]

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...?

 

[HallsofIvy]

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.

 

[Iyafrady]

Hmm, the book says one possible answer is a=-2,b=c=1

 

[Rainbow Child]

What is the exact statement of the problem from your book? Does it says anything about the vectors A,B, independent or something else?

 

[Iyafrady]

what i posted is the EXACT statement, its from a vector analysis course.

 

[Rainbow Child]

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.

 

[jostpuur]

I hope nobody succeeded seeing my previous post, which I deleted quickly. I cannot believe how frozen my brains were...

 

[Iyafrady]

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!

 

[berkeman]

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...