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Thread: Linear System

  1. #1
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    Linear 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 $\displaystyle AX=0$.
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  2. #2
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    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 $\displaystyle AX=0\rightarrow X=A^{-1}0=0$
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