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Let A = [[1, 1], [1, 3], [−2, −3]] and b = [b1, b2, b3]. For Ax = b to be solvable, which one of the following options is the correct condition on b1, b2, and b3:

1. b1 + b2 + b3 = 1
2. 3b1 + b2 + 2b3 = 0
3. b1 + 3b2 + b3 = 2
4. b1 + b2 + b3 = 2

Correct answer: 3b1 + b2 + 2b3 = 0

Solution

The correct option is based on the requirement for the system of equations represented by Ax = b to have a solution. This condition arises from the need for the vector b to lie in the column space of matrix A, which is determined by the linear combinations of its columns. The equation 3b1 + b2 + 2b3 = 0 ensures that b can be expressed as a linear combination of the columns of A.

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