# Exam

Exam

1.- Which is the rank of this matrix:

 a) 1 b) 2 c) 3 d) 4

2.- Which is the rank of this matrix:

 a) 1 b) 2 c) 3 d) 4

3.- Discuss the rank of this matrix depending on the value of m:

 a) rk = 3 if m ≠ 2; rk = 2 if m = 2 b) rk = 2 if m ≠ 2; rk = 3 if m = 2 c) rk = 3 if m ≠ 1; rk = 2 if m = 1 d) rk = 3 if m ≠ -2; rk = 2 if m = -2

4.- Discuss this system by using the Rouché-Fröbenius Theorem:

 a) Independent system b) Dependent system c) Inconsistent system d) Independent system; x = y = z = 0

5.- Discuss this system by using the Rouché-Fröbenius Theorem:

 a) If k = 1 inconsistent system; if k ≠ 1 independent system b) If k = -1 inconsistent system; if k ≠ -1 independent system c) If k = 3 dependent system; if k ≠ 3 independent system d) Independent system k € R

6.- Discuss this system by using the Rouché-Fröbenius Theorem:

 a) If k = 1 dependent system; if k ≠ 1 independent system b) If k = -1 inconsistent system; if k ≠ -1 independent system c) If k = -1 dependent system; if k = 1 inconsistent system; if k € R-{-1,1} independent system d) If k = -1 inconsistent system; if k = 1 dependent system; if k € R-{-1,1} independent system

7.- Discuss this system by using the Rouché-Fröbenius Theorem:

 a) If a = 2 and b = 1 dependent system; if a = 2 and b ≠ 1 inconsistent system; if a ≠ 2 independent system b) If a = 2 and b = 1 inconsistent system; if a = 2 and b ≠ 1 dependent system; if a ≠ 2 independent system c) If a = 2 inconsistent system; if a ≠ 2 independent system d) If a = 2 and b = -1 dependent system; if a = 2 and b ≠ -1 inconsistent system; if a ≠ 2 independent system

8.- Solve by using Cramer's rule:

 a) x = y = z = 0 b) x = y = z = λ; λ € R c) x = 2λ; y = z = λ; λ € R d) x = z = λ; y = -λ; λ € R

9.- Solve by using Cramer's rule:

 a) x = z = λ; y = -7λ; λ € R b) x = 27/23; y = -17/46; z = 9/46 c) x = -3; y = 4; z = 0 d) x  = 2; y = 1; z = -2

10.- Solve by using Cramer's rule for the values of k that make the system consistent:

 a) x = -k; y = k/2; z = 2k/7 b) x = (1-k2)/(k2+1); y = -(k2+k)/(k2+1); z = k2/(k2+1) c) x = (1-k2)/(k2+1); y = (k2+k)/(k2+1); z = -k2/(k2+1) d) x = 7; y = k2 + k; z = (k+1)/(k2+1)