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If the determinant of [[a1 + b1 x, a1 x + b1, c1], [a2 + b2 x, a2 x + b2, c2], [a3 + b3 x, a3 x + b3, c3]] equals 0, which of the following is/are possible conditions?

1. x = 1 for all a_i, b_i
2. x = -1 for all a_i, b_i
3. the determinant of [[a1, b1, c1], [a2, b2, c2], [a3, b3, c3]] = 0
4. x = +/-2 for all a_i, b_i

Correct answer: the determinant of [[a1, b1, c1], [a2, b2, c2], [a3, b3, c3]] = 0

Solution

The determinant factors to (1 - x²)*det[[a_i],[b_i],[c_i]], so it vanishes when x = +/-1 or when det[[a1,b1,c1],...] = 0; the condition guaranteed for all a_i, b_i is that this latter determinant is zero (the x = +/-1 options only work in conjunction). The always-sufficient condition is det[[a_i, b_i, c_i]] = 0.

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