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 $\displaystyle AX=0$.
Follow Math Help Forum on Facebook and Google+
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 $\displaystyle AX=0\rightarrow X=A^{-1}0=0$
View Tag Cloud