Solution
= Solution
\b[Group the matrix entries by columns.] Each column of $BA$ is $B$ times the corresponding column of $A$. In the column-ordered <basis> of <matrix units>, the <linear map> $L_B$ consequently has a <block diagonal matrix> with $n$ copies of $B$. The <determinant> of a <block diagonal matrix> is the product of its block <determinants>, giving
$$
\boxed{\det L_B=(\det B)^n.}
$$
This argument does not require $B$ to be an <invertible matrix>.