Suppose $\displaystyle A$ is a square matrix and the homogeneous system $\displaystyle (A^2)^T X = 0$ has a unique solution.Can any conclusion be made on the linear system $\displaystyle AX=0$.

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- Feb 5th 2010, 12:16 AMproblemLinear System
Suppose $\displaystyle A$ is a square matrix and the homogeneous system $\displaystyle (A^2)^T X = 0$ has a unique solution.Can any conclusion be made on the linear system $\displaystyle AX=0$.

- Feb 5th 2010, 03:04 AMmath2009
Yes, $\displaystyle AX=0$ has unique solution

Proof :

$\displaystyle rank(A)=rank(A^T)$ ref attachment

$\displaystyle n\geq rank(A)\geq rank(A^2)=rank((A^2)^T)=n\rightarrow rank(A)=n$

So $\displaystyle AX=0\rightarrow X=A^{-1}0=0$