Question

Linear Algebra,

Can someone help me prove that [(A)^T]^-1 = (A)^-1]^T

Answer 1

Let's assume it's false for the sake of contradiction:

\[(A^\top)^{-1} \ne (A^{-1})^\top\]
We can multiply both sides by \(A^\top\) to get:
\[A^\top(A^\top)^{-1} \ne A^\top(A^{-1})^\top\]
Then recognize that the left hand side is a matrix multiplied by its inverse, so it's equal to the identity:
\[I \ne A^\top(A^{-1})^\top\]
We know that there is this transpose rule for matrices:
\[(BC)^\top = C^\top B^\top\] so we use that on the right hand side to get:
\[I \ne (A^{-1} A)^\top\]
But that's just equal to the identity,
\[I \ne I^\top\]
but this statement is false, so we've come to a contradiction and it must be true!