<s>[INST] Solve $A^nx=b$ for an idempotent matrix Let $$A=\begin{bmatrix}2& 3& -4\\ 0& 1 & 0\\ 0.5& 1.5 &-1\end{bmatrix},~ b=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}.$$ Show that $A$ is idempotent and solve the matrix equation $$A^nx=b$$ for each positive integer $n$.
 [/INST] Is there a specific part where you're having trouble?
For the first part, do exactly as the question asks -- multiply $A$ by itself and check that you get $A$ back.
For the second part, if $A^2=A$, then $A^n = \ldots$?
EDIT: You can solve the system using Gaussian elimination. Starting from
$$\left[\begin{array}{ccc}2 & 3 & -4\\0 & 1 & 0\\1/2 & 3/2 & -1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}1\\0\\0\end{array}\right]$$
we can add $-4$ times the last row to the first to get
$$\left[\begin{array}{ccc}0 & -3 & 0\\0 & 1 & 0\\1/2 & 3/2 & -1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}1\\0\\0\end{array}\right],$$
and now adding 3 times the second row to the first gives
$$\left[\begin{array}{ccc}0 & 0 & 0\\0 & 1 & 0\\1/2 & 3/2 & -1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}1\\0\\0\end{array}\right]$$
and the system has no solution.
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