Inverse and Transpose of Lorentz Transformation I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise.
By definition:

*

*$(\Lambda^T)^{\mu}{}_{\nu} = \Lambda_{\nu}{}^{\mu}$

*$\Lambda^T \eta \Lambda = \eta$, which is $\Lambda^{\rho}{}_{\mu} \eta_{\rho \sigma} \Lambda^{\sigma}{}_{\nu} = \eta_{\mu \nu}$ in index notation.

We can further manipulate the second definition (as done in Tong's lecture notes):
$\begin{align}
\Lambda^{\rho}{}_{\mu} \eta_{\rho \sigma} \Lambda^{\sigma}{}_{\nu} &= \eta_{\mu \nu} \\
\Lambda^{\rho}{}_{\mu} \Lambda_{\rho \nu} &= \eta_{\mu \nu} \\ 
\Lambda^{\rho}{}_{\mu} \Lambda_{\rho \nu} \eta^{\nu \sigma} &= \eta_{\mu \nu} \eta^{\nu \sigma} \\ \Lambda^{\rho}{}_{\mu} \Lambda_{\rho}{}^{\sigma} &= \delta_{\mu}^{\sigma} \\  \Lambda_{\rho}{}^{\sigma} \Lambda^{\rho}{}_{\mu} &= \delta_{\mu}^{\sigma}
\end{align}$
Recalling that $({\Lambda^{-1}})^{\sigma}{}_{\rho}$ is defined through:
$$({\Lambda^{-1}})^{\sigma}{}_{\rho} \Lambda^{\rho}{}_{\mu} = \delta^{\sigma}_{\mu}$$
This then implies that $$({\Lambda^{-1}})^{\sigma}{}_{\rho} = \Lambda_{\rho}{}^{\sigma}\tag{A}$$ but according to definition 1, doesn't $$({\Lambda^{T}})^{\sigma}{}_{\rho} = \Lambda_{\rho}{}^{\sigma}~?\tag{B}$$ This seems to incorrectly imply that $$({\Lambda^{T}})^{\sigma}{}_{\rho} = ({\Lambda^{-1}})^{\sigma}{}_{\rho}\tag{C}.$$ I'm not really sure what step of my logic is incorrect.
Tong makes the following comment on the (A) result:

The result is analogous to the statement that the inverse of a rotation matrix is the transpose matrix. For general Lorentz transformations, we learn that the inverse is sort of the transpose where “sort of” means that there are minus signs from raising and lowering. The placement of indices in tells us where those minus signs go.

This comment seems to suggest that (B) is incorrect - although it just seems like mere application of definition 1.
Edit to clarify question after initial responses:
From this analysis, why is it incorrect to conclude that $$\Lambda^{-1} = \Lambda^T~?\tag{D}$$ We know this matrix equation is not true, but why is this not implied by $({\Lambda^{T}})^{\sigma}{}_{\rho} = \Lambda_{\rho}{}^{\sigma} = ({\Lambda^{-1}})^{\sigma}{}_{\rho}$ since the indices in $\Lambda^{T}$ and $\Lambda^{-1}$ are the same?
Further Clarification Question:
Some of the responses will reveal that in fact only the matrix equation D is incorrect because the index structure of $\Lambda$ is $\Lambda^{\mu}{}_{\nu}$, the index structure of $\Lambda^{-1}$ is ${(\Lambda^{-1})}^{\mu}{}_{\nu}$, but the index structure of $\Lambda^T$ is ${(\Lambda^{T})}_{\mu}{}^{\nu}$ (not ${(\Lambda^{T})}^{\mu}{}_{\nu}$).
However, this leaves one final question: how can we show explicitly that the matrix $\Lambda^T$ should correspond to this different index structure? Using this structure does make everything consistent again, but how does this follow from defining the matrix $\Lambda$ as corresponding to $\Lambda^{\mu}{}_{\nu}$?
 A: $\def\th{\theta}
\def\sp{\hspace{1ex}} 
\def\b{\beta} 
\def\a{\alpha} 
\def\m{\mu} 
\def\n{\nu} 
\def\g{\gamma}
\def\d{\delta} 
\def\mt{\eta} 
\def\mti{\mt^{-1}}
\def\F{\Phi} 
\def\Ft{\widetilde{\F}}
\def\L{\Lambda}
\def\Li{\L^{-1}}
\def\Lt{\L^T}
\def\id{\mathbb{I}}$From the invariant interval one derives
\begin{align*}
\L^\b_{\sp\a} \mt_{\b\g} \L^\g_{\sp\d} 
&= \mt_{\a\d}.\tag{1}
\end{align*}
Let $\F_\a^{\sp\b}=\L^\b_{\sp\a}$, so (1) is written as
\begin{align*}
\F_\a^{\sp\b} \mt_{\b\g} \L^\g_{\sp\d} 
&= \mt_{\a\d}\tag{2}
\end{align*}
with matrix interpretation
\begin{align*}
\F\mt\L=\mt.\tag{3}
\end{align*}
By index gymnastics (2) is massaged into the form
$$\F^\a_{\sp\g} \L^\g_{\sp\d} = \d^\a_\d$$
so
$\F^\a_{\sp\b} = (\Li)^\a_{\sp\b}$.
Note that, critically, the first index has been raised and the last lowered.
Letting $\Ft$ be the matrix determined by $\F^\a_{\sp\b}$ we have
$$\Ft=\Li.$$
We are, however, interested in $\F_\a^{\sp\b}$.
We find
$\F_\a^{\sp\b} 
= \mt_{\g\a}\F^\g_{\sp\d}\mt^{\b\d}
= \mt_{\g\a}(\Li)^\g_{\sp\d}\mt^{\b\d}$.
This has the matrix interpretation
\begin{align*}
\F &= \mt\Li\mti. \tag{4}
\end{align*}
In fact, (4) follows immediately from (3), illustrating the usefulness of the matrix representation.
The confusion boils down to one between $\F$ and $\Ft$.
It is straightforward to show from (4) and the general form for $\L$ that $\F=\Lt$. (See comment below.)
Thus,
$$\Lt\mt\L=\mt$$
is the correct matrix representation of (1).
Let us illustrate the difference between $\F$ and $\Ft$ with a specific nontrivial example.
Represent $\L^\a_{\sp\b}$ by
$$\L = \left[
\begin{array}{cccc}
 \gamma  & 0 & 0 & -\beta  \gamma  \\
 0 & \cos \theta  & -\sin \theta  & 0 \\
 0 & \sin \theta  & \cos \theta  & 0 \\
 -\beta  \gamma  & 0 & 0 & \gamma  \\
\end{array}
\right].$$
Then
$$\F = \Lt = \left[
\begin{array}{cccc}
 \gamma  & 0 & 0 & -\beta  \gamma  \\
 0 & \cos \theta  & \sin \theta  & 0 \\
 0 & -\sin \theta  & \cos \theta  & 0 \\
 -\beta  \gamma  & 0 & 0 & \gamma  \\
\end{array}
\right].$$
But
$$\Ft = \Li = \left[
\begin{array}{cccc}
 \gamma  & 0 & 0 & \beta  \gamma  \\
 0 & \cos \theta  & \sin \theta  & 0 \\
 0 & -\sin \theta  & \cos \theta  & 0 \\
 \beta  \gamma  & 0 & 0 & \gamma  \\
\end{array}
\right].$$
(This is a boost in the $z$ direction and rotation about the $z$-axis.)
Comment
When one studies the general solutions to (1) one finds that they are combinations of rotations and boosts. Note that for a rotation $\mt \L(\th)^{-1}\mti=\mt\L(-\th)\mti=\L(-\th)=\L(\th)^T$ and that for a boost, $\mt \L(\b)^{-1}\mt=\mt \L(-\b)\mti= \L(\b)=\L(\b)^T$.
A: *

*Note that the conventional definition of transposed matrix
$$(\Lambda^T)_{\nu}{}^{\mu} ~:=~ \Lambda^{\mu}{}_{\nu}.\tag{1'} $$
is slightly different than OP's definition (1).


*In plain English: When we don't apply the metric, the Lorentz matrix $\Lambda$ has conventionally NW-SE slanted indices, while the transposed matrix $\Lambda^T$ has SW-NE slanted indices.
See also e.g. this & this related Phys.SE post.


*Incidentally, OP's eq. (1) is consistent with eq. (1') after appropriately raising and lowing indices with the metric.


*OP's eq. (D) is wrong because it doesn't comply with above convention.


*More details: In matrix form OP's eqs. (A)-(C) read
$$ \Lambda^{-1}~=~ (\eta\Lambda\eta^{-1})^T,\tag{A} $$
$$ \eta^{-1}\Lambda^T\eta~=~ (\eta\Lambda\eta^{-1})^T,\tag{B} $$
$$  \eta^{-1}\Lambda^T\eta~=~\Lambda^{-1},\tag{C} $$
respectively, which are indeed all true. Eq. (C) follows from the definition
$$ \Lambda^T\eta\Lambda~=~\eta$$
of a Lorentz matrix.
A: I think the reason this is confusing is that tensor and matrix notations are being mixed up in ways that do not actually make sense. Also, the notation fails to observe that the Lorentz transformation takes one coordinate system to another. Usually one would need a prime on either $\mu$ or $\nu$ (one talks of primed and unprimed coordinates). The Lorentz transformation is a particular instance of a general coordinate transformation, which can be written
$$k^{\mu'}_\nu = x^{\mu'}_{,\nu} = \frac{\partial x^{\mu'} }{\partial x^{\nu}}$$
In such a case ${\mu'}$ runs over rows and ${\nu}$ runs over columns. It does not matter which index is "first" (I definitely prefer accounts like that of Dirac, General Theory of Relativity which explicitly does not put either index first in this case). The transpose would swap covariant and contravariant indices, which kind of makes a nonsense of your definition 1. Transpose is used for matrices, because the ordering of indices is important for matrix multiplication. But in general relativity this is already taken care of in index notation through Einstein's summation convention. I don't recall any of my preferred texts for gtr using transpose, but I have to confess, if an author did use it, I think I would rapidly find another author.
A: The problem here is that your definition (1) is incorrect if the metric isn't the identity in your coordinates. The only correct way of raising/lowering indices is by contraction with the metric.
A simple proof that your definition can't be right: Suppose our metric is $\def\d{{\rm d}}\d s^2={\d x^2\over4}-\d v^2$, and $\Lambda_\mu{}^\sigma=\begin{bmatrix}5\over4&3\over2\\3\over8&5\over4\end{bmatrix}$.
Then your definition gives $\Lambda^\mu{}_\sigma=\begin{bmatrix}5\over4&3\over8\\3\over2&5\over4\end{bmatrix}$, while mine gives $\Lambda^\mu{}_\sigma=\begin{bmatrix}5\over4&-{3\over2}\\-{3\over8}&5\over4\end{bmatrix}$.
Then if you use your definition to evaluate $\Lambda^\rho{}_\mu\eta_{\rho\sigma}\Lambda^\sigma{}_\nu$, you get $\begin{bmatrix}-{119\over64}&-{225\over128}\\-{225\over128}&-{391\over256}\end{bmatrix}$; if you use mine, you get $\begin{bmatrix}1\over4&0\\0&-1\end{bmatrix}$.
A: After a very helpful discussion in the comments section and reading the responses, I thought I would type up (from my perspective) what I have learnt in case it will help anyone with the same question.
$$({\Lambda^{T}})^{\sigma}{}_{\rho} = \Lambda_{\rho}{}^{\sigma} = ({\Lambda^{-1}})^{\sigma}{}_{\rho}$$
is in fact a correct statement, but we have to be careful when converting this back into a matrix equation.
We should interpret $\Lambda$ as $\Lambda^{\mu}{}_{\nu}$, $\Lambda^{-1}$ as $({\Lambda^{-1}})^{\mu}{}_{\nu}$, but $\Lambda^T$ should be interpreted as $(\Lambda^T)_{\mu}{}^{\nu}$.
Therefore, we cannot interpret $({\Lambda^{T}})^{\sigma}{}_{\rho}$ as $\Lambda^T$ so equation D is incorrect. Instead, using the metric, $({\Lambda^{T}})^{\sigma}{}_{\rho} = \eta^{\sigma \alpha} (\Lambda^T)_{\alpha}{}^{\beta} \eta_{\beta \rho}$. So, instead of matrix equation D, we really should have:
$$\Lambda^{-1} = \eta \Lambda^T \eta$$
