# nontrivial solution for a homogeneous system

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• January 26th 2011, 07:53 PM
superdude
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.
• January 26th 2011, 08:21 PM
harish21
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 $x = A^{-1} \times 0 = 0$.