What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?

makeupwn

makeupwn

Answered question

2022-08-06

What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?
I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc>0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU>0. I would love to see a solution along these goemetrical lines.

Answer & Explanation

Chaya Garza

Chaya Garza

Beginner2022-08-07Added 10 answers

Explanation:
The probability that the determinant is 0 is 0. Thus by symmetry the required probability is 1 2
pleitatsj1

pleitatsj1

Beginner2022-08-08Added 4 answers

Step 1
The products ad and bc have identical distributions, and a d b c is the convolution of the density distributions of ad and b c, which is symmetrical and centered at zero. So the probability is 1/2.
Step 2
Edit: that under the hypothesis that a,b,c,d have the same density distribution.

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