# Thread: I have 2 conflicting flashcards. Which one is wrong?

1. ## I have 2 conflicting flashcards. Which one is wrong?

Card #1

Card #2

I remember being told that the trivial solution is simple solution where all variables are zero. However the second card says I will have a trivial solution if the determinant is non-zero which means that the variables have unique numbers that are not just zeros. Is there something I'm missing?

2. ## Re: I have 2 conflicting flashcards. Which one is wrong?

both are correct.

the first is a condition on the input. Clearly (hopefully) any linear combination of zeros produces zero.

the second is a condition on the system. Certain systems have greater than zero dimensional null spaces, i.e. sets of non-zero inputs that produce a zero output.

a way to identify these systems is via a determinant of zero.

3. ## Re: I have 2 conflicting flashcards. Which one is wrong?

Where did you get "determinant is non-zero which means that the variables have unique numbers that are not just zeros"? The determinant being non-zero tells you nothing about the values of the variables. With a matrix equation of the form Ax= b, where A is a known n by n matrix, x is an unknown n vector, and b is a known n vector then
1) If det(A) is non-zero, there exist unique vector x satisfying the equation. If b is the 0 vector that x vector is the 0 vector.

2) If det(A) is 0 and b is not the 0 vector then there is no solution.

3) If det(A) is 0 and b is the 0 vector then there are an infinite number of solutions.

I see no contradiction between then two cards.