(c) Is it possible for an n × n matrix A to satisfy A3 = In without A being invertible? Explain

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17-10-2019
Mathematics

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(c) Is it possible for an n × n matrix A to satisfy A3 = In without A being invertible? Explain

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LammettHash LammettHash · Answer 1 · 2019-10-17T17:49:21+02:00

LammettHash LammettHash

17-10-2019

If [tex]A^3=I[/tex], then [tex]A^3=AA^2=I[/tex] which means [tex]A[/tex] and [tex]A^2[/tex] must be inverses of one another, so [tex](A^2)^{-1}=A[/tex] and [tex]A^{-1}=A^2[/tex], so [tex]A[/tex] must be invertible.

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