Question

Which of the following cases may lead to non trivial solutions in case of system of linear equations according to Cramer's rule Convention?

• A. D1 = 0, D2 = 0, D3 = 0
• B. D1 = 0, D2 = 0, D3 != 0
• C. D1 != 0, D2 != 0, D3 = 0
• D. D1 != 0, D2 != 0, D3 != 0

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

D1 = 0, D2 = 0, D3 = 0

B

D1 = 0, D2 = 0, D3 != 0

C

D1 != 0, D2 != 0, D3 = 0

D

D1 != 0, D2 != 0, D3 != 0

1) D != 0, at least one of D1,D2,D3 != 0

Then there exists a non trivial solution implies nontrivial solution possible

2) D ! = 0, D1 = D2 = D3 = 0,

only the trivial solution exists implies nontrivial solutions are impossible

3) D = 0, D1 = D2 = D3 = 0,

either infinite or no solutions implies nontrivial solution may be possible

4) D = 0, at least one of D1,D2,D3 != 0

no solutions possible implies no nontrivial solutions possible

From case 1 and 3 non trivial solutions are possible if at least one of D1,D2,D3 != 0 or if D1 = D2 = D3 = 0 I.e all the options may give nontrivial solutions depending on what value D has. So all options are correct.