Calculate multiplication of a cipher matrix and a plain matrix There is a matrix $A$ that we don't want to be revealed, and there is a matrix $B$ that is known. Is there any way to calculate $A \times B$ without revealing $A$?
$A$ can be encrypted. But we don't want anyone to know it, but we want anyone to be able to calculate the multiplication of it to another matrix.
Is there anyway to do such a thing?
Yes there is. It is called functional encryption for inner product (IPFE), and there are many schemes for that, with different features in terms of underlying assumption (typically DDH, DCR, or LWE), security (IND-CPA, IND-CCA, static or adaptive, etc) and compactness.
An IPFE has four algorithms:

$\mathsf{Setup}$ generates a public encryption key $\mathsf{pk}$ and a master secret key $\mathsf{msk}$
$\mathsf{Enc}$ takes as input a fixed-length message vector $\vec m$, and outputs a ciphertext $c$
$\mathsf{KeyDer}$ takes as input a linear function $f$, the master secret key $\mathsf{msk}$, and outputs a functional secret key $\mathsf{sk}_f$
$\mathsf{Decrypt}$ takes as input a functional secret key $\mathsf{sk}_f$, a ciphertext $c$, and outputs $f(\vec m)$, where $\vec m$ is the message encrypted in $c$.

In your case, the linear function $f$ would be the function $f_B: A \mapsto A\times B$.
IPFE was first defined in this paper, where it was also constructed from a variety of standard assumptions. Since this work, there has been many follow ups, achieving stronger security notions, or more features.
In your context: the matrix $A$ would be encrypted with $\mathsf{Enc}$, and the dealer would reveal the secret key $\mathsf{sk}_B$ associated to the function $f_B: A \mapsto A \times B$. This would allow anyone to obtain $A\times B$ from any ciphertext encrypting a matrix $A$.
Of course, usual safeguards apply: if your matrix $B$ is invertible, this would just reveal $A$ in clear, so $B$ better be a highly compressing, or very low rank matrix, if you want this process to preserve some security.