1.- Which is the rank of this matrix:
a) 1
b) 2
c) 3
d) 4
2.- Which is the rank of this matrix:
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
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)
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