Question

Prove that if A is a square matrix then (A^T)^-1=(A^-1)^T

Answer 1

\[(A^{T})^{-1}=(A^{-1})^{T}\]

Answer 2

since A is invertible, it can be broken up into the product of two matrices \(A=XY\) combine the properties \((AB)^T=B^TA^T\) and \((AB)^{-1}=B^{-1}A^{-1}\)

Answer 3

This is what I wrote down, does this make sense haha \[CA^{T}=A^{T}C=I\] \[(A^{-1})^{T}A^{T}=(A(A^{-1}))^{T}=(AA^{-1})^{T}=I^{T} = I\]
and
\[A^{T}(A^{-1})^{T}=((A^{-1})A)^{T}=(A^{-1}A)^{T}= I^{T} = I\]

Answer 4

so they both equal I

Answer 5

I guess that last part is tautological :/

Answer 6

hey, I like your way better!