2
$\begingroup$

I am to prove or find a counterexample for the following problem: Let $A, B, C, D, F \in GL(n, \mathbb{R})$. If $D^{-1}(A+B+C)D=F$ is correct then $A=F-B-C$ is correct.

I have not found a counterexample though there might be one - I don't know. This is how far I get:

Let $E=A+B+C$. Then the problem can be expressed as "If $D^{-1}ED=F$ is correct then $E=F$ is correct." Also, $D^{-1}ED=F \Leftrightarrow ED=DF$

But I have no clue how I could show $ED=DF \Rightarrow E=F$.

  • 0
    I$t$ is easy to cook up counter e$x$amples once you know what's going on. Do read about matrix similarity (or matrix diagonalization). It will make this very clear.2012-11-17

2 Answers 2

2

Hint: $E=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ should work as a counterexample.

  • 0
    You're welcome.2012-11-17
2

Hint: Try simple $2\times 2$ counterexamples. There is also counterexample where $B=C=0$.

  • 1
    Then try $B=-C=I_n$.2012-11-17