Recent content by geoffrey159

Thank you for your help !

I think I get it, but it is a little difficult: ## \begin{align*} \det B &= \sum_{\sigma\in S_n} \epsilon(\sigma) b_{\sigma(1)1}\cdots b_{\sigma(n)n}\\ &= \sum_{\sigma\in S_n} \epsilon(\sigma) a_{\sigma(1)1}\cdots a_{\sigma(n)n} (-1)^{\#S_\sigma} \end{align*}## where set ##S_\sigma =...

I have received an explanation in terms of permutations, but I didn't get it. That's why I tried to find another way. What do you mean ?

The problem with the determinants in the cofactors is that neighbor columns have the same sign and break the 'chessboard' structure. That's why I reverse the sign of the ##k-1## first columns, in order to recover this structure, which allows me to apply ##{\cal P}(n-1)##.

Homework Statement Given a matrix ##A = (a_{ij})##, we define matrix ##B = \begin{pmatrix} a_{11} & - a_{12} & a_{13} & \cdots \\ - a_{21} & a_{22} & -a_{23} & \cdots \\ a_{31} & - a_{32} & a_{33} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{pmatrix}##. Another way to define ##B## is...

Homework Statement [/B] Given a general triangle ABC, find the geometric locus of points such that the three orthoprojection onto the sides of the triangle are aligned. Homework Equations Let's call A', B', and C' the orthoprojection of a given point M onto (AB) , (BC) , and (AC). M satisfies...

Hi, here is the picture

Homework Statement [/B] Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity. I think I have understood the problem, could you verify my demo please ? Thanks Homework Equations The Attempt at...

Hehehe that's funny lol. But more seriously, wasps can be deadly to allergic people.

What are your stories/experiences with venomous creatures ?

Oh I know the culprit, it was a masouran (or plotosus lineatus), which is a sort of small catfish. It is cute and peaceful, but its sting is very painful

I spent my holidays in Mauritius island (indian ocean) and was fishing with cousins living here when I got stung several times by a poisonous fish in the foot. I felt a fast growing pain, was sweating and feeling dizzy, and by the time I reached my cousins on the beach, I started to have...

That's unfair :-)

Number 10, the infinite product is finite and equal to ##\pi / 2##. For this I wrote ##\prod_{n = 1 }^N \frac{4n^2}{4n^2-1} = 4^N (N!)^2 \frac{2^N N!}{(2N)!} \frac{2^N N!}{(2N+1)!} = \frac{4^{2N} (N!)^4}{(2N+1) ((2N)!)^2 } ## And Stirling's formula leads to the conclusion

The chief of the forty thieves has to devise a strategy that will give him the highest possible probability of survival. Let us call strategy ##(\alpha,\beta)## the strategy that consists in placing ##\alpha## white and ##\beta## black balls in box 1, and let us define the events ##A##...