%% Linear Algebra

%% System of Equations
% In linear algebra a system of equations can be represented by a
% coefficient matrix. Matlab can solve all types of linear equations
% automatically. We will demonstrate how Matlab matrix manipulation
% commands can be used to perform row operations on a system of linear
% equations. 
%
% Consider the system:
%
% $$4 x + 3 y + z = 10$$
%
% $$5 x + 4 y + 2 z = 4$$
%
% $$x -y +z = 15$$
%

%% Coefficient Matrix
% We define a standard coefficient matrix.
A = [4 3 1; 5 4 2; 1 -1 1]

%% Right-hand side values
% These are the values from the right hand side of the equations
C = [10;4;15]

%% Augmented Matrix
% Row reduction is done on the augmented matrix.
AA = [A C]

%% Swap Rows 1 and 3
% The syntax manipulates two rows of the matrix at the same time by
% indexing them in a matrix. The colon indicates all columns. 
AA([1 3], :) = AA([3 1], :)

%% Eliminate values in rows 2 and 3
% We use a trick to manipulate two rows at the same time. The final term in
% the command is a 2x1 matrix multiplied by a 1x4 matrix. This produces two
% copies of the first row, each scaled differently.
AA([2 3],:) = AA([2 3],:) - [5;4]*AA(1,:)

%% Divide through second row
AA(2,:) = AA(2,:) ./ 9

%% Row Operation
AA(3,:) = AA(3,:) - 7 * AA(2,:)

%% Divide through third row
AA(3,:) = AA(3,:) ./ -0.6667

%% Add rows
AA(1,:) = AA(1,:) + AA(2,:)

%% Add scaled rows
AA([1 2], :) = AA([1 2], :) + [-0.6667; 0.3333] * AA(3,:)

%% Matlab can computer Determinants automatically
% The det command will compute a determinant.
det(A)

%% Matlab can calculate the Row Reduced Echelon Form
% A much simpler approach is to just ask Matlab to compute the row reduced
% echelon form of the augmented matrix.
rref(AA)

%% Matrix Division
% Matlab supports a form of matrix division. In the equation:
%
% $$ AX = C $$
%
% Solving for X gives:
%
% $$ X = A^{-1}C $$
%
% Matlab can perform this calculation with "left division":
A\C