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Which of the following matrices cannot be expressed as the square of a 3 × 3 matrix with real elements?

1. [1 0 0; 0 1 0; 0 0 1]
2. [1 0 0; 0 -1 0; 0 0 -1]
3. [-1 0 0; 0 -1 0; 0 0 -1]
4. [1 0 0; 0 1 0; 0 0 -1]

Correct answer: [-1 0 0; 0 -1 0; 0 0 -1]

Solution

A 3 × 3 matrix with real elements cannot have a negative determinant when squared, so the matrix with a negative determinant on the diagonal cannot be expressed as the square of a 3 × 3 matrix with real elements.

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