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Let A be the 2x2 matrix with first row (0, 1) and second row (0, 0), and let I be the 2x2 identity matrix. If (I + A)²⁰ - 19A = [[alpha, beta], [gamma, delta]], find alpha + beta + gamma + delta.

1. 0
2. 1
3. 2
4. 3

Correct answer: 3

Solution

Since A² = 0, by the binomial theorem (I+A)²⁰ = I + 20A. Then (I+A)²⁰ - 19A = I + 20A - 19A = I + A = [[1,1],[0,1]]. Sum of entries: 1 + 1 + 0 + 1 = 3.

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