Linear algebra help

[granite osprey]

i'm supposed to check if this is true or false but im a bit confused on how i should go bout this question

[fast sinew]

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[sacred timber]

What do you think it is ?

[granite osprey]

its false

but im not sure how im supposed to prove it

bcs technically like

[sacred timber]

Find a counter example

A pretty simple one

Like take a diagonal matrice for instance

[granite osprey]

wise words

chat did i cook

[sacred timber]

Not the simplest example but it works

[granite osprey]

im back

what the frick does it even mean

[sacred timber]

A being skew symmetric and B being any square matrice of same size then there is some non zero X such ABX=0

Why do you think that is ?

[fast sinew]

@granite osprey

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[fast sinew]

@sacred timber The user still needs help with this help request.

[granite osprey]

oh wait sorry i didnt mean to ping

[granite osprey]

ok so wait

nontrivial means that theres other solutions possible right

so like

all values are leading ones?

[sacred timber]

If C is a square matrice then Cx=0 has non trivial solutions if and only if C is not invertible

[granite osprey]

OHH

ok

so were asking if AB is invertrible?

[sacred timber]

Yes

What is the determinant of a skew symmetric matrice in odd dimensions ?

[granite osprey]

zero

so

its not invertible?

[sacred timber]

Yes, why is that ?

[granite osprey]

uhhhhhh

bcs it is what it is

[sacred timber]

det(A^T)=det(A)

A^T is the transpose of A

[granite osprey]

yes

but when its skew symmetric

its A^t = -A

no?

[sacred timber]

yes

and what is det(-A) ?

In terms of det(A)

[granite osprey]

ITS

WAIT

[sacred timber]

I’ll wait

Dw

[granite osprey]

cause det(kA) = k^n det(A)

so in that case

[sacred timber]

Yea

if A is n x n

[granite osprey]

det(-A) is (-1)^n det(A)

[sacred timber]

yes and here n is odd

So it’s -det(A)

so det(A)=-det(A)

ie det(A)=0

[sacred timber]

A not being invertible implies AB isn’t

So you can conclude

[granite osprey]

so theres nontrivial solution?

[sacred timber]

Yes

[granite osprey]

WOOO

u explain so well what the helly

thank u

[sacred timber]

Np