find 2×2 matrix such that $ A^2 =-I$ . I think that eigenvalue of $A^2$ is 0 or positive but eigenvalue of $-I$ is negative so we haven't matrix wih this condition.
2026-03-29 13:22:44.1774790564
find 2×2 matrix such that $ A^2 =-I$
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Given condition implies $A^2+I=0$. So characteristic polynomial is $x^2+1$. So any matrix with trace 0 and determinant 1 will satisfies your condition!
In particular, take $$A=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}$$