Math Is Fun Forum

If you take the square matrix:

It's inverse is

Doesn't sound very interesting!

What about this?

consider this matrix

Now if

means the rth row and cth column element.

now delete the rth row and cth column to make a n-1 X n-1 matrix find it's determinent and call it 

in the below matrix

is represented by

form the matrix

(the plus/minus sign is becuase you need to know the size of the matrix to know the sign of the bottom row)

In words what has been done:

change lowercase to capital (or from E to D)

start with the outer two elements which form diagonals swap the elements round (there's only one element in each so nothing changes)

go to the next diagonal pair (one inwards on either side) and swap them (the elements within the diagonal see above) and keep going

also the sign of the outermost diagonals stay the same the next pair change sign, whether the trailing diagonal (what all these diagonals are parallel to) changes depends on the size.

Calculate the determinent Δ in the following way:

(multiply the first row of original matrix by the first column of the new matrix)

To find the detiminent of the n-1 X n-1 matrices use the above method (which means looking at n-2 X n-2 matrices and so on down to 2 X 2 whose determinent is above

if you put say that the determinent of a 1 X 1 matrix is its one element (which means the method above dosen't work on 1 X 1)

then for the 2 X 2:

then right:

and sustitute

which fits the above

using this method you can find the inverse of a n X n matrix provided you know how to do the n-1 X n-1 (you just have to work your way backwards to the 2 X 2

unfortunatly I don't know how to prove it!

I've recently been doing a search for these (A polychoron is a 4D polytope made up of polyhedra, like a polyhedra 3D made up of polygona)

The properties are:

1) Strictly-convex from body to cell (this is generalising it to higher dimensions)

2) Non-vertex transitive

3) Regular faced

I found 193 of these so far and hope to find more

They're based on the Johnson Solids

I've been trying to figure out why this shape is not uniform. All it's vertices have 3 squares and 1 triangle (if I've got that right)

"as there are pairs of vertices such that there is no isometry of the solid which maps one into the other. Essentially, two types of vertices can be distinguished by their "neighbors of neighbors."

This is the only clue I have but I don't know what it means. Can someone explain it to me?:D I would be most grateful.