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Let A = [1 0 0; 0 1 0; a b −1] and let I denote the 3 × 3 identity matrix. Then the value of A² + 2A⁴ + 4A⁶ is

1. 7A⁸
2. 7A⁷
3. 8I
4. 6I

Correct answer: 7A⁸

Solution

Direct multiplication gives A^2 = I, so A^4=A^6=I and A^2+2A^4+4A^6=7I. Since 7A^8 = 7(A^2)^4 = 7I, the answer is 7A^8.

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