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Alize Kuhn

Feb 20, 2025

Squared Matrix to get a negative identity matrix?

There are 2 × 2 matrices A such that A^2 = −I2. For example, if A = (matrix)0 −11 0 , then A2 = −I2. Is there a 3 × 3 matrix A such that A^2 = −I3? If so, find such an A. If not, show no such A exists.(hint: use the fact that if X and Y are n × n matrices then |XY | = |X||Y |)
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|-I2| = 1.A 2x2 matrix A that satisfies |A|^2 = |-I2| must have |A| = +/- 1, as the example does. |-I3| = -1.A 3x3 matrix A that satisfies |A|^2 = -1 is impossible.(however, there are matrices such that A^3 = -I3)... Show More
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