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For a matrix A satisfying A² = 0, what is the expansion of (aI + bA)²?

1. (aI + bA)² = a² I + b² A² + 2ab AI
2. (aI + bA)² = a² I + b² A² + 2ab A
3. (aI + bA)² = a² I + b² A + 2ab AI
4. (aI + bA)² = a² I + b² A² + ab A

Correct answer: (aI + bA)² = a² I + b² A² + 2ab A

Solution

Expand: (aI+bA)^2 = a^2 I + ab(IA) + ab(AI) + b^2 A^2 = a^2 I + 2ab A + b^2 A^2. Since A^2 = 0, this equals a^2 I + 2ab A, i.e. a^2 I + b^2 A^2 + 2ab A.

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