Is-this-statement-true-or-not-A-M-3-R-tr-A-0-and-A-2-t-A-I-3-

Question Number 143774 by TheHoneyCat last updated on 18/Jun/21

Is this statement t r u e or not ?

\exists A \in M_{3} (R) ∣ tr (A) = 0 and A^{2} +^{t} A = I_{3}

Commented by TheHoneyCat last updated on 18/Jun/21

Oh, I actually found the answer. �� sorry. Should I send it?

Commented by ArielVyny last updated on 18/Jun/21

y e s

Answered by TheHoneyCat last updated on 19/Jun/21

let A be a matrix such that : A^{2} +^{t} A = I_{3}

-^{t} A = A^{2} - I_{3}

so - A =^{t} A^{2} - I_{3}

thus - A = (A^{2} - I_{3}) - I_{3}

\Leftrightarrow 0_{M_{3} (R)} = A^{2} - 2 A + I_{3} - I_{3} + A

\Leftrightarrow 0_{M_{3} (R)} = A^{2} - A

in a word : the polynomial (X - 1) X cancels

therefore, let μ_{A} be the m i n i m a l p o l y n o m i a l of A

μ_{A} \in {X, X - 1, (X - 1) X}

μ_{A} = X \Rightarrow A = 0_{M_{3} (R)} but 0^{2} +^{t} 0 \neq I_{3}

μ_{A} = X - 1 \Rightarrow A = I_{3} \Rightarrow tr (A) = 3 \neq 0

μ_{A} = (X - 1) X \Rightarrow A is d i a g o n a l i z a b l e with 1 and 0 as e i g e n v a l u e s

\Rightarrow \exists G \in G l_{3} (R) ∣ G A G^{- 1} \in {(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}), (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})}

\Rightarrow tr (A) \in {2, 1}

\Rightarrow tr (A) \neq 0

So the answere is n o, there are no such matrices