Question

if A is a matrix then prove that (A^t)^-1 = (A^-1)^t if A is a matrix then prove that (A^t)^-1 = (A^-1)^t @Mathematics

Answer 1

Remember first that \( (AB)^T = A^T B^T \) and \( (A^T)^T = A \). Now \( A^T (A^T)^{-1} = I \ \) hence taking the transpose of both sides we have
\[ (A^T)^T ((A^T)^{-1})^T = I^T = I \]
But \( (A^T)^T = A \) hence \( A ((A^T)^{-1})^T = I \) which means that
\[ ((A^T)^{-1})^T = A^{-1} \]
Now take the transpose of both sides
\[ (A^T)^{-1} = (A^{-1})^T \]

Answer 2

(AB)^t = B^t A^t isn't it

Answer 3

Sorry yes. But the rest of the argument holds up. Just flip the order of the matrices where I use that.

Answer 4

Note to self: do not write mathematics before coffee.

Answer 5

thank u

Answer 6

how can u say A^t (A^t)^-1=I