Tensor product of irreducible representation and one-dimensional representation is irreducible

Statement

Suppose $G$ is a group and $K$ is a field. Suppose $\rho_1:G \to GL(V_1)$ is an [[irreducible linear representation] of $G$ over $K$ and $\rho_2: G \to GL(V_2)$ is a one-dimensional linear representation of $G$ over $K$. Then, the tensor product of linear representations $\rho_1 \otimes \rho_2$ is an irreducible linear representation of $G$ over $K$. Also, this tensor product is projectively equivalent to $\rho_1$.