clc
echo on
% Example 3
%
% Basic Matrix Operations
%
% Matrix Creation

A = [1 2; 4 3]

size(A)

A(2,1)

B = [ 1  3
      0 -1 ]

% Scalar Multiplication
% multiply each element in the matrix by the scalar.
%

2*A

(-3)*B

pause
clc
% Matrix Addition
% The matrices must be the same size,
% then add corresponding elements.
%

A = [1 2 3; 1 0 -1]

B = [1 1 1; 3 2 1]

A+B

pause
clc
% Linear Combinations
% Combines scalar multiplication and matrix addition
%

A = [ 1 2
      4 3 ]

B = [ 1  3
      0 -1 ]

2*A - 3*B

pause

% Linear Combinations of Vectors
%

x = [1; 2; 3]

y = [1 0 -1]'

2*x - 3*y
pause

% Matrix Multiplication
%
A = [1 2; 4 3]

B = [1 0 1; 0 -1 2]

A*B

% Inner Product
%
x' * y

% Outer Product
%
x * y'

pause
clc
% Solving a system of linear equations
%

A = [3 2 -1; 1 -1 0; 0 5 1]

y = [5; 6; 1]

% Solve A*x = y
%
x = A\y

format short e
residual = A*x - y

pause
% The LU factorization
%
[L, U, P] = lu(A)

pause
clc
% This factorization always exists
% Even if the system can't be solved
%
C = [1 2 3; 1 0 -1; 1 1 1]

[L, U, P] = lu(C)

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