Question

Help please !!

Answer 1

Let A and B in Mn(R) such that :
A.B = B.A and
Show that if \[p \in \mathbb{N}*\] then \[\det (A ^{p}+B ^{p})\ge0\]

Answer 2

\[Mn(R) = M _{n}(\mathbb{R})\]

Answer 3

@eliassaab ana idea please.

Answer 4

@eliassaab

Answer 5

@lgbasallote

Answer 6

@Hero

Answer 7

what topic is this? maybe i can tell you the right person to tag

Answer 8

@lgbasallote : it's linear algebra : Matrices

Answer 9

i guess @Chlorophyll can helpyou here

Answer 10

@dpaInc

Answer 11

I think it's in relation with matrix reduction!!

Answer 12

Let
\[
A=\left(
\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}
\right)\\
B=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right)
\]

Then AB=BA=0
\[
A^3=\left(
\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}
\right)\\
B^3=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right)\\

A^3 + B^3=A^3
\\
Det(A^3 + b^3) =Det(A^3)=-1



\]

The above is a counter example if I understood your question correctly.

Answer 13

no , because there is another condition on A and B , it's \[\det(A+B)\ge0\] And in you example det(A+B)=-1

Answer 14

excuse me Mr elias I forgot it in the question ''and.....''

Answer 15

So It's clear?

Answer 16

I do not know. I can not check. But I met our math teacher this morning and he gave me a solution.So logically there will be a mistake in your example (maybe).

Answer 17

but surely there is error in your example. As for integers p pair, we have :
\[\det ( A ^{2p}+B ^{2p})=\det((A ^{p}+iB ^{p})\times(A ^{p}-iB ^{p}))=\det(A ^{p}+iB ^{p})\times \det(A ^{p}-iB ^{p})\]

Answer 18

There might be some truncation errors. I will double check.