Let $\displaystyle A=B=\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ 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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- Jan 26th 2011, 07:53 PMsuperdudenontrivial solution for a homogeneous system
Let $\displaystyle A=B=\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ 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. - Jan 26th 2011, 08:21 PMharish21
If $\displaystyle ad-bc \neq 0$ 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 $\displaystyle A^{-1}B$. In the case of a homogeneous system Ax=0, the vector b is 0 and the system has only the trivial solution $\displaystyle x = A^{-1} \times 0 = 0$.