Linear System

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February 5th 2010, 12:16 AM
problem

Linear System

Suppose $A$ is a square matrix and the homogeneous system $(A^2)^T X = 0$ has a unique solution.Can any conclusion be made on the linear system $AX=0$ .
February 5th 2010, 03:04 AM
math2009

1 Attachment(s)

Yes, $AX=0$ has unique solution

Proof :
$rank(A)=rank(A^T)$ ref attachment
$n\geq rank(A)\geq rank(A^2)=rank((A^2)^T)=n\rightarrow rank(A)=n$
So $AX=0\rightarrow X=A^{-1}0=0$

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