Rouché-Fröbenius Theorem

Let the system A·X = B, where:

Then:

–(1) The system is consistent ↔ rk(A) = rk(A*)

–(2) If rk(A) = rk(A*) = r →

Example 1:

Example 2:

Example 3:

Exercise: Discuss the following systems by using the Theorem:

Solutions:

a) Independent system

b) Inconsistent system if a = -1; dependent system if a = 2; independent system if a € R-{-1,2}

c) Independent system if m ≠ -1; dependent system if m = -1