1 - Solve the system .
The augmented matrix A|b of the system is: ,
which is row equivalent to the reduced row-echelon matrix B|c: .
The solution of the system is easily found: . The other unknowns are completely arbitrary. This system has n=6, m=4, r=3, and so ∞^{6-3}=∞^{3} solutions.
2 - Solve the system , where t is a real number.
The augmented matrix and its reduced form are: . The system is consistent if and only if t=2. In this case the reduced row-echelon form of the matrix is: . We read off the unique solution of the system: . From a geometrical point of view this result can be interpreted as follows: given two non parallel straight lines and a third variable straight line, find the value of t for which this third line has a unique point in common with the preceding two.
3 - Solve the system , where s and t are real numbers.
Write the augmented matrix A|b: . Now start applying Gauss-Jordan algorithm. ; .
Now we must distinguish between to cases: