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Math Help - nontrivial solution for a homogeneous system

  1. #1
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    nontrivial solution for a homogeneous system

    Let A=B=\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix} Show that the homogeneous sytem Ax=0 has only the trivial solution iff ad-bc≠0

    I know it has to do with there not being a column of 0s in A.
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  2. #2
    MHF Contributor harish21's Avatar
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    If ad-bc \neq 0 the matrix A invertible and the the system Ax=b is consistent for every choice of the column vector b and the unique solution is given by A^{-1}B. In the case of a homogeneous system Ax=0, the vector b is 0 and the system has only the trivial solution x = A^{-1} \times 0 = 0.
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