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Math Help - system of equations / conditions question

  1. #1
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    system of equations / conditions question

    system of equations:-
    note :- | - (denotes matrix bracket) beta & alpha are real

    |1 2| |x| |α|
    |1 β| |y| = |1|

    under what conditions of beta and alpha give the following results:-
    1) unique solution
    2)infinite number of solutions
    3) no solution

    thanks
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  2. #2
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    1) A unique solution exists only when the rank of the matrix equals the row/column dimension. So when the columns are linearly independent.
    2) When the columms are linearly dependent.
    3) When the system of equations generate a faulty value for x and/or y.
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  3. #3
    Junior Member
    Joined
    Oct 2006
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    Quote Originally Posted by acu04385 View Post
    system of equations:-
    note :- | - (denotes matrix bracket) beta & alpha are real

    |1 2| |x| |α|
    |1 β| |y| = |1|

    under what conditions of beta and alpha give the following results:-
    1) unique solution
    2)infinite number of solutions
    3) no solution

    thanks
    The first step is to reduce the system. In this case it's nearly immediate. At that point you can clearly see how to set a and b
    to satisfy 1-3. What follows are just my suggestions.

    3) set b so that the column space is a 1-dimensional subspace of R^2, and set a so that the RHS sits outside of the column space.
    Consider b = 2 and a = 0.

    2) set b like 3) above, and set a so that the RHS sits in the column space.
    Consider b = 2 and a = 1.

    1) set b so that the column space is all of R^2, and set a to anything you like.
    Consider b = 3.
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