Determine the matrix $B$ such that the following equation $L$ is a linear transformation from $R^{n \times n}$ to $R^{n \times n}$

Question

Let $A$ and $B$ be given $n \times n $. Determine the matrix $B$ such that the following equation $L$ is a linear transformation from $R^{n \times n}$ to $R^{n \times n}$

$$ L(X)=A(B+BX)+X^tB, \forall X \in R^{n \times n}$$

Where $t$ is the transpose

how to do this

because we to show that this linear transformation means $L(X+Y)=L(X)+L(Y), L(aX)=aL(x)$

Answer 1

$L$ is a linear transformation if and only if $AB=0$ (where $0$ stands for the zero matrix). Just note that $X \to ABX$ and and $X \to X^{t} B$ are both linear. Hence L is linear if and only if the constant map $X \to AB$ is linear which is true only when the constant (matrix) AB is $0$.