# Show that if M is invertible, M^t is also invertible

• Aug 15th 2010, 01:25 PM
experiment00
Show that if M is invertible, M^t is also invertible
hmm, how do i prove this one?

plus,

(M^t)^-1 = (M^-1)^t
• Aug 15th 2010, 01:33 PM
yeKciM
Quote:

Originally Posted by experiment00
hmm, how do i prove this one?

plus,

(M^t)^-1 = (M^-1)^t

from
$A^T(A^{-1})^T=(A^{-1}A)^T=I^T = I$
we conclude that it's really
$(A^{-1})^T=(A^T)^{-1}$
• Aug 15th 2010, 02:40 PM
math2009
if M is invertible, then $\det(M)\neq 0$

$\because~\det(M^T)=\det(M)\neq 0$, so $M^T$ is invertible