How Do the E and B Fields Transform Through Rotation and Boost?

[Matterwave]

Homework Statement

So, I'm asked to find how the fields (E, and B) transform by transforming the electromagnetic field tensor.

The transformations are a) rotation around y axis, and b) boost along z.

Homework Equations

$$F'_{\mu\nu}=\Lambda^\mu_\rho \Lambda^\nu_\sigma F_{\rho\sigma}$$

The Attempt at a Solution

So, doing part a) I think the above equation should be right...and for the transformation matrices, I just put the rotation around y:

$$\Lambda^\mu_\nu =\begin{bmatrix} 1&0&0&0 \\ 0&cos(\theta)&0&sin(\theta)\\0&0&1&0\\0&-sin(\theta)&0&cos(\theta) \end{bmatrix}$$

I trid just doing the matrix multiplication twice, but that can't be right because the answer I get is not anti-symmetric (and so I can't extract the information I need). Does the equation above not represent matrix multiplying twice? I thought it did since the transformation for a 4-vector is simply matrix multiplication once.

The answer does not seem to be just:

$$F'_{\mu\nu}=\begin{bmatrix} 1&0&0&0 \\ 0&cos(\theta)&0&sin(\theta)\\0&0&1&0\\0&-sin(\theta)&0&cos(\theta) \end{bmatrix}(\begin{bmatrix} 1&0&0&0 \\ 0&cos(\theta)&0&sin(\theta)\\0&0&1&0\\0&-sin(\theta)&0&cos(\theta) \end{bmatrix} \begin{bmatrix} 0&E_x&E_y&E_z \\ -E_x&0&-B_z&B_y\\-E_y&B_z&0&-B_x\\-E_z&-B_y&B_x&0 \end{bmatrix})$$

 

[Matterwave]

Oh, my friend helped me and figured out I need to sandwich the F inside the lambdas...and one of them has to be inverse...but which one should be inverse for me transforming a covariant tensor? Is it Lambda^-1 F Lambda or Lambda F Lambda^-1?

I think the only difference is that I rotate either by +theta or -theta...but supposing I want to rotate by +theta...?