% with diary we produce a log of all commands being
% typed in the MATLAB command prompt window
diary off; % close the buffer
% with diary on we continue logging
A = [0 1 2
3 4 5
6 7 7]

A =

     0     1     2
     3     4     5
     6     7     7

det(A)

ans =

     3

% We define a righthand side vector b
% if we want to solve A * x = b
b = [1 1 1]'

b =

     1
     1
     1

x = A\b

x =

    -1
     1
     0

r = b - A*x

r =

     0
     0
     0

% The residual r shows that
% x is a solution to A*x =b
who

Your variables are:

A    b    x    
ans  r    

% To save the data for later use
% we use "save" command
cd 'C:/Users/janv/Documents'
save('mydata','A','b','x','r')
% let us clear the data
clear
A;
{??? Undefined function or variable 'A'.
} 
load 'mydata'
% With the file session.txt we see
% the commands that we typed in.
% To store data we must use save/load.
% using an LU factorization solves
% a linear system more efficiently
[L,U,P] = lu(A)

L =

    1.0000         0         0
         0    1.0000         0
    0.5000    0.5000    1.0000


U =

    6.0000    7.0000    7.0000
         0    1.0000    2.0000
         0         0    0.5000


P =

     0     0     1
     1     0     0
     0     1     0

% A is decomposed as a product
% of L (lower triangular) with U
% (upper triangular), with permutations in P.
P*A

ans =

     6     7     7
     0     1     2
     3     4     5

L*U

ans =

     6     7     7
     0     1     2
     3     4     5

% we see P*A = L*U
% By default, the output format of 
% floats is with 4 digits after the
% decimal point.
format long e
L

L =

  Column 1

    1.000000000000000e+000
                         0
    5.000000000000000e-001

  Column 2

                         0
    1.000000000000000e+000
    5.000000000000000e-001

  Column 3

                         0
                         0
    1.000000000000000e+000

r

r =

     0
     0
     0

format short; % go back to the default
L

L =

    1.0000         0         0
         0    1.0000         0
    0.5000    0.5000    1.0000

% Geometrically, that x is a solution
% of A*x = b, means that we can write
% b as a linear combination of the
% columns of the matrix A.
% We can selects the columns of A:
c1 = A(:,1)

c1 =

     0
     3
     6

c2 = A(:,2); c3 = A(:,3);
c1*x(1) + c2*x(2) + c3*x(3)

ans =

     1
     1
     1

x

x =

    -1
     1
     0

% We see that the third column of A
% does not play a role.
AA = A(:,1:2)

AA =

     0     1
     3     4
     6     7

% Now we can look at the linear system
% AA*y = b.
y = AA\b

y =

   -1.0000
    1.0000

format long e
y

y =

   -1.000000000000002e+000
    1.000000000000002e+000

% With the backslash operator,
% we solve a linear system in the
% least squares sense.
[Q,R] = qr(A)

Q =

  Column 1

                         0
   -4.472135954999580e-001
   -8.944271909999160e-001

  Column 2

    9.128709291752774e-001
    3.651483716701102e-001
   -1.825741858350548e-001

  Column 3

    4.082482904638624e-001
   -8.164965809277263e-001
    4.082482904638632e-001


R =

  Column 1

   -6.708203932499369e+000
                         0
                         0

  Column 2

   -8.049844718999244e+000
    1.095445115010334e+000
                         0

  Column 3

   -8.497058314499203e+000
    2.373464415855722e+000
   -4.082482904638640e-001


format short 
Q

Q =

         0    0.9129    0.4082
   -0.4472    0.3651   -0.8165
   -0.8944   -0.1826    0.4082

R

R =

   -6.7082   -8.0498   -8.4971
         0    1.0954    2.3735
         0         0   -0.4082

[Q,R] = qr(AA)

Q =

         0    0.9129    0.4082
   -0.4472    0.3651   -0.8165
   -0.8944   -0.1826    0.4082


R =

   -6.7082   -8.0498
         0    1.0954
         0         0

% The QR decomposition writes A
% as a product of Q (orthogonal)
% and R (upper triangular).
Q'*Q

ans =

    1.0000   -0.0000   -0.0000
   -0.0000    1.0000    0.0000
   -0.0000    0.0000    1.0000

% The inverse of Q is its transpose Q'.
Q*R

ans =

         0    1.0000
    3.0000    4.0000
    6.0000    7.0000

% Via the QR decomposition, solving
% AA*y = b now becomes solving 
% (Q*R)*y = b or R*y = Q'*b.
bb = Q'*b;

yy = R\bb

yy =

   -1.0000
    1.0000

diary off