- #1

- 333

- 47

**Theorem**

If A is non singular then ##(A^T)^{-1} = (A^{-1})^T##

**Proof**

The first part of proof shows that the inverse unambiguously decided. Then

##A^T(A^T)^{-1} = I##

and

##I = A^{-1}A = (A^{-1}A)^T = A^T(A^{-1})^T##

Where the second step is possible because ##I = I^T##. From the equations above

##A^T(A^T)^{-1} = A^T(A^{-1})^T \Longrightarrow (A^T)^{-1} = (A^{-1})^T##

The text claims the last step is possible thanks to the inverse being unabigously decided. Why does this allow us to use cancellation?