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Square Matrix Scenarios

  1. Mar 6, 2012 #1
    Hi everyone,

    I've been reading through my textbook for some guidance on this, but I have yet to find anything. The question asks to "describe the set of possible solutions when the following occur (assuming Ax=b). I've written my own knowledge in italics - however, I think they require an answer in terms of "no solution"/"# of unique solutions" or "infinite solutions".

    (i) detA = 0, b = 0 (no inverse)
    (ii) detA ≠ 0, b = 0 (inverse possible)
    (iii) detA = 0, b ≠ 0 (no inverse)
    (iv) detA ≠ 0, b ≠ 0 (inverse possible)

    The way I learnt was that if there was 3 solutions, there had to be 3 variables/pieces of information, otherwise it was redundant or inconsistent depending on what you were given. I've yet to learn using determinants and the b-vector.

    Any insight would be helpful - Thanks.
     
  2. jcsd
  3. Mar 6, 2012 #2

    Mark44

    Staff: Mentor

    It's true that the matrix A doesn't have an inverse, but they're asking about the equation Ax = 0. Does this equation have a) no solutions, b) exactly 1 solutions, c) an infinite number of solutions?
    If det(A) ≠ 0, then A definitely has an inverse, so how many solutions does the equation Ax = 0 have?
    Again, how many solutions does the equation Ax = b have?
    As before, if det(A) ≠ 0, then A definitely has an inverse, so how many solutions does the equation Ax = b have?
    If you have 3 equations and 3 unknowns, such a system might have no solutions, 1 solution, or an infinite number of solutions. It can never have exactly three solutions. As a really simple example, here is a very simple system of 3 equation in 3 variables:
    x = 2
    y = 3
    z = 5
    This can be represented as a matrix equation in the form Ax = b, where A is
    [tex]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

    and b is
    [tex]\begin{bmatrix} 2 \\ 3 \\ 5 \end{bmatrix}[/tex]

    This system has one solution, not three; namely x = 2, y = 3, z = 5.

     
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