If A is an invertible matrix,
is A^t always invertible? If t is a fractional exponent, would it guarantee invertibility?
2026-03-26 06:17:53.1774505873
Counterexamples Invertibility
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Let $t=p/q$.
Let $B = A^t$, i.e. $B^q = A^p$. Then, $B^q (A^{-1})^p = A^p A^{-p} = I_n$, so $B(B^{q-1}A^{-p}) = I_n$, so $B$ is invertible, and $B^{q-1}A^{-p}$ is its inverse.
Caveat: in general, $A^{p/q}$ is not defined; if it is, it is not uniquely defined.