Pure Maths---System of linear equation

ai) det (E) != 0
=> k != 1 and 0 <- this is and nor or. (think about it)
aii) put k=0 in E and use guassian elimination.

aii) put k =1 in E and use guassian elimination, give, gamma = - alpha+2 beta

bi) if F has non-zero solution, then det[F]=0
by notice that F is transpose of E]
which k = 1 or 0

assumed E is consistent. for k = 0 or 1

[a b c].[ alpha beta gamma] != 0

=> [a b c ] E [ x y z ] !=0 ; for [x y z ] is the solution of E, which, we assumed E is consistent
=> [a b c ] F^T [x y z] !=0
=> [ F [a b c] ] ^T [x y z] !=0
=> 0 [x y z] != 0
=> 0 != 0

thus the assumption fail. i.e E is not consistent

bii)
observer that if E consistent, the sign of beta and gamma are the same.
by constructing different sign of them, we can make E is not consistence

for k = 1
[a b c] =[ 1 -2 1]
[ alpha beta gamma] = [ 0 1 -2]

for k = 0
[a b c] = [2 1 -1]
[ alpha beta gamma] = [ 0 -1 1]]

either gives E is inconsistent.