%% Inverses of Matrices
% This demo shows how to visualize matrices as images and uses this to
% illustrate matrix inversion.
%
% Copyright 1984-2002 The MathWorks, Inc. 
% $Revision: 5.14 $

%%
% This is a graphic representation of a random matrix. The RAND command creates
% the matrix, and the IMAGESC command plots an image of the matrix,
% automatically scaling the color map appropriately.

n = 100;
a = rand(n);
imagesc(a);
colormap(hot);
axis square;

%%
% This is a representation of the inverse of that matrix. While the numbers in
% the previous matrix were completely random, the elements in this matrix are
% anything BUT random. In fact, each element in this matrix ("b") depends on
% every one of the ten thousand elements in the previous matrix ("a").

b = inv(a);
imagesc(b);
axis square;

%%
% But how do we know for sure if this is really the correct inverse for the
% original matrix? Multiply the two together and see if the result is correct,
% because just as 3*(1/3) = 1, so must a*inv(a) = I, the identity matrix. The
% identity matrix (almost always designated by I) is like an enormous number
% one. It is completely made up of zeros, except for ones running along the main
% diagonal.

%%
% This is the product of the matrix with its inverse: sure enough, it has the
% distinctive look of the identity matrix, with a band of ones streaming down
% the main diagonal, surrounded by a sea of zeros.

imagesc(a*b);
axis square;