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Let A be the 3x3 matrix with rows [0, 0, -1], [0, -1, 0], [-1, 0, 0]. Which of the following statements is true?
- A⁻¹ does not exist
- A² = I
- A = 0 (zero matrix)
- A = (-1)*I
Correct answer: A² = I
Solution
The determinant of A equals 1 (non-zero), so A is invertible and A⁻¹ exists. Direct multiplication shows A² = I, making A its own inverse. A is not the zero matrix nor a scalar multiple of I (the diagonal entries of A are 0, -1, 0, not all equal).
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