QR decomposition using Givens rotations:
some random matrix A:
   0.840    0.394    0.783 
   0.798    0.912    0.198 
   0.335    0.768    0.278 
   0.554    0.477    0.629 
matrix Q:
   0.633   -0.578    0.156 
   0.601    0.314   -0.712 
   0.252    0.752    0.439 
   0.417   -0.031    0.525 
matrix R: (should be right-triangular)
   1.328    1.191    0.947 
   0.000    0.621   -0.201 
   0.000    0.000    0.433 
matrix QR: (should be equal A):
   0.840    0.394    0.783 
   0.798    0.912    0.198 
   0.335    0.768    0.278 
   0.554    0.477    0.629 
matrix Q^T*Q: (should be equal 1)
   1.000    0.000    0.000 
   0.000    1.000   -0.000 
   0.000   -0.000    1.000 

Linear System: 
random square matrix A:
   0.365    0.513    0.952    0.916 
   0.636    0.717    0.142    0.607 
   0.016    0.243    0.137    0.804 
   0.157    0.401    0.130    0.109 
random right-hand side b :      0.999      0.218      0.513      0.839 
solution x to Ax=b       :      -2.94       3.06      0.892     -0.378 
check: Ax: =>b?          :      0.999      0.218      0.513      0.839 

Inverse matrix: 
inverse matrix A^-1:
   0.386    2.266   -1.671   -3.535 
  -0.539   -0.718    0.601    4.105 
   1.249   -0.772   -0.883    0.317 
  -0.058    0.303    1.247   -1.222 
check: matrix A*A^-1 =>1? :
   1.000   -0.000    0.000    0.000 
  -0.000    1.000   -0.000    0.000 
   0.000   -0.000    1.000    0.000 
  -0.000   -0.000    0.000    1.000 
check: matrix A^-1*A =>1? :
   1.000    0.000    0.000   -0.000 
  -0.000    1.000   -0.000    0.000 
   0.000    0.000    1.000    0.000 
   0.000   -0.000    0.000    1.000