# Matrix Proof

• Jan 23rd 2013, 08:19 AM
guybrush92
Matrix Proof
Hey, I've been trying to prove that the inverse of the transpose of a matrix, is equal to the transpose of the inverse of a matrix, i.e.

\$\displaystyle (A^T)^(-1)=(A^(-1))^T\$

(also, some help on how to use TEX would be useful...)
• Jan 23rd 2013, 08:40 AM
BobP
Re: Matrix Proof
Are you allowed to assume that \$\displaystyle (AB)^{T}=B^{T}A^{T}\$ ?

If so, take the transpose of \$\displaystyle AA^{-1}=I\$ and postmultiply by \$\displaystyle (A^{T})^{-1}.\$

For the LaTex input, use the Go Advanced option and 'press' the \$\displaystyle \Sigma\$ button and between the two [Tex] brackets type the line in as

(A^{T})^{(-1)}=(A^{(-1)})^{T}.

The ^ symbol translates as 'to the power of', and the power is then closed within the curley brackets.
• Jan 23rd 2013, 09:54 AM
Deveno
Re: Matrix Proof
by definition an inverse for an nxn matrix A is an nxn matrix B such that AB = I.

and....

\$\displaystyle A^T(A^{-1})^T = (A^{-1}A)^T = I^T = I\$