I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam.(adsbygoogle = window.adsbygoogle || []).push({});

So my stab at it:

det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0

If A is invertible, there exists a B such that

AB = BA = I

det(AB) = det(I)

detA detB = 1

Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't

Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.

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# Simple Linear Algebra (determinant invertibility)

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