Fix matrices A, B € S4 , where B is in addition invertible. Prove that there exists an invertible symmetric matrix CE S" satisfying C2 = B . [Hint: Build C from an eigenvalue decomposition of A by taking the square roots of the eigenvalues (Why can you take square roots of eigenvalues?). ]

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Q: Prove that if 0 is an eigenvalue of A, then A is not an invertible matrix.

A: Please see the proof below.