# Solving matrix equation $A^n x = b$ [duplicate]

Possible Duplicate:
Solve $A^nx=b$ for an idempotent matrix

It is given that:

$$A = \begin{bmatrix} 2 & 3 & -4 \\ 0 & 1 & 0 \\ 1/2 & 3/2 & -1 \end{bmatrix}$$

and

$$b = \begin{bmatrix} 1 & 0 & 0 \\ \end{bmatrix}$$

Solve $A^n x = b$ for each positive integer n.

$A$ is idempotent so $A^n$ for positive integers of $n$ will still be equal to $A$. So the equation simply becomes $Ax = b$.

However, unless I am mistaken, $A$ is not invertible so I cannot solve for $x$ by multiplying both sides of the equation by $A^{-1}$.

What should I do instead?

## marked as duplicate by N. S., Bombyx mori, user53153, Austin Mohr, PaulJan 28 '13 at 3:55

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• Is $A$ idempotent? It does not seem so! – Maesumi Jan 28 '13 at 3:28

## 2 Answers

Hint: Look at the second equation, it gives the value of $x_2$. Now use substituition.

If you find $Ax=b$ with $b=[1,0,0]$, then the system has no solution. The reason is the first row is a linear combination of second and third row, so if both of them are $0$ then the first one must be $0$ as well.