Im asked to solve for X given the equation $$ (A^{-1}X)^{-1} = (AB^{-1})^{-1}(AB^2) $$ What I have done so far is: $$ ((A^{-1}X)^{-1})^{-1} = ((AB^{-1})^{-1}(AB^2))^{-1} \\A^{-1}X = ((AB^{-1})^{-1}(AB^2))^{-1} \\A^{-1}X = AB^{-1}A^{-1}B^{-2} \\AA^{-1}X = AAB^{-1}A^{-1}B^{-2} \\X=A^2B^{-1}A^{-1}B^{-2} $$ Is this correct or did I preform something that is not true? Thank you.
2026-04-07 12:55:46.1775566546
matrix algebra with invertibles
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Your third step is wrong. Note that $(XY)^{-1}=Y^{-1}X^{-1}$
$\\A^{-1}X = ((AB^{-1})^{-1}(AB^2))^{-1} \\A^{-1}X = B^{-2}A^{-1}AB^{-1} \\A^{-1}X = B^{-3} \\X=AB^{-3} $