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 $\rho _{1}$.