"B=A^{-1}"
Let "A^*" be adjoint of "A",
"B^*" be adjoint of "B".
1)
"A^{-1}=\\frac{A^*}{detA}" and "B^{-1}=\\frac{B^*}{detB}"
"B=A^{-1}=\\frac{A^*}{detA}" (multyply by "B^{-1}" )
"BB^{-1}=\\frac{A^*}{detA}B^{-1}"
"E=\\frac{A^*}{detA}\\frac{B*}{detB}"
"E=\\frac{A^*}{detA}\\frac{B*}{det(A^{-1})}"
"E=A^*B^*"
"B^*=(A^*)^{-1}"
2)
"A=B^{-1}"
"A^T=(B^{-1})^T"
"A^T=(B^T)^{-1}"
Another proof:
"AB=E"
"(AB)^T=E^T"
"B^TA^T=E"
"(B^T)^{-1}B^TA^T=(B^T)^{-1}E"
"A^T=(B^T)^{-1}"
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