# Homework Help: Prove that Ax=Ix has only the trivial solution

1. Oct 6, 2012

### drawar

1. The problem statement, all variables and given/known data
Let A be a square matrix of order n such that $2{A^2} + A = 4I$. Prove that the only $x \in ℝ^n$ that satisfies $Ax = Ix$ is x=0.

2. Relevant equations
$Ax = 0$ has only the trivial solution iff A is invertible.

3. The attempt at a solution
The problem would be pretty trivial if the given equation was Ax=0, but how am I gonna tackle it when the RHS is Ix? TIA!

2. Oct 6, 2012

### tiny-tim

hi drawar!

hint: if Ax = x, what is A2x ?

3. Oct 6, 2012

### drawar

It seems your hint works out quite nicely, please check my working:
Observe that $Ax = Ix = x \Rightarrow {A^2}x = Ax = x$
Hence $2{A^2} + A = 4I \Leftrightarrow 2{A^2}x + Ax = 4Ix \Leftrightarrow 2x + x = 4x \Leftrightarrow x = 0$
This completes the proof.

4. Oct 6, 2012

fine!

5. Oct 6, 2012

### drawar

Yeah, thanks!