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For the matrices A = [[3, -4],[1, -1]] and B = [[-29, 49],[-13, 18]], if (A^15 + B) [[x],[y]] = [[0],[0]], then among the following which one is true? (1) x = 5, y = 7 (2) x = 18, y = 11 (3) x = 11, y = 2 (4) x = 16, y = 3

1. x = 5, y = 7
2. x = 18, y = 11
3. x = 11, y = 2
4. x = 16, y = 3

Correct answer: x = 11, y = 2

Solution

The equation (A^15 + B) [[x],[y]] = [[0],[0]] implies that the vector [[x],[y]] is in the null space of the matrix (A^15 + B). The correct option, x = 11 and y = 2, satisfies this condition, indicating that this vector results in the zero vector when multiplied by (A^15 + B).

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