help with a group solution

Algebra

help with a group solution

Is the solution group of the system A^3X = 0
, Is equal to the solution group of the system AX = 0

If this is true you will prove it, if not give a counterexample.

thank you.
Guest

Re: help with a group solution

Not necessarily. It depends on A. If A is an "invertible" operator then we could apply the inverse to both sides, $$A^{-2}(A^3x)= (A^{-2}A^3)x= Ax= 0$$ to show that x must be 0. But if A is not invertible, then $$A^3x$$ can be 0 when Ax is not necessarily 0. For a counter example, consider $$A= \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}$$. $$A^3$$ is the 0 matrix so $$A^3x= 0$$ for all x.

HallsofIvy

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