Setting Up a Matrix with Order Unity Elements: A Scientist's Guide

[Ashley1nOnly]

Homework Statement

A determinant with all elements of order unity may be surprisingly small. The Gilbert determinant Hij=(I+j-1)^-1, i,j=2... n is notorious for its small values.

Homework Equations

The Attempt at a Solution

I just need help setting up the matrix and I can solve it myself. Thanks

 

[Ashley1nOnly]

I know how to find the numbers for the matrix but I'm confused as to how big the matrix is supposed to be. Am I doing a 1 by 1 then a 2 by 2 then a 3 by3 or am I doing a 3 by 3

 

[TJGilb]

Your notation is a little unclear to me. I assume i represents the matrix row and j represents the matrix column. If that equation is $H_{ij} = (i+j-1)^{-1}$ with dimension equal to 2, then the value of each element is given by this equation with whichever row and column number plugged in.

 

[TJGilb]

I know how to find the numbers for the matrix but I'm confused as to how big the matrix is supposed to be. Am I doing a 1 by 1 then a 2 by 2 then a 3 by3 or am I doing a 3 by 3

Your problem statement doesn't give any direction for actually solving anything. Is there more to it?

 

[Ashley1nOnly]

Yes sorry. I'm stuck on the part where it says by order of n=1,2, and 3. What is is asking me to do?. I have already found my matrix using the given equation( it repeats itself in a decreasing way)

 

[Ashley1nOnly]

Calculate the value of the Hoover determinant of order n for n=1,2, and 3.

Is the problem

 

[TJGilb]

If it tells you to solve for an order of 1, 2, and 3, then it's telling you to compute the determinant of a 1x1, 2x2, and 3x3 matrix.

 

[Ashley1nOnly]

Quick question. When you are finding the determinant of a matrix. Do the rows and columns have to be equal?

 

[TJGilb]

Quick question. When you are finding the determinant of a matrix. Do the rows and columns have to be equal?

Yes. You can only take the determinant of a square matrix.

 

[Ashley1nOnly]

For my answers I got
1
1/12
And I am working on the third

 

[Ashley1nOnly]

For the 3 by 3
0.00046297

 

[TJGilb]

That looks right!