Let \(X_n,n\geq 1\), be a sequence of independent identically distributed random matrices with values in \(L(R^n)\) such that \(\Pr\{X_n=A\}=p\), \(\Pr\{X_n=B\}=q=1-p\), \(0<p<1\), and let \(Y_n=X_n X_{n-1}\cdots X_1\). Denote by \(\chi(Y)=\lim_{n\to\infty}n^{-1} \|Y_n\|\pmod P\) the index of exponential growth that is used for the investigation of asymptotics of \(Y_n\). It is proved that for idempotent matrices \(A\) and \(B\) the following relation holds true: \(\chi(Y)=pq\max_{\lambda\in S(BA)} \ln \lambda\), where \(S(BA)\) is the set of eigenvalues of \(BA\).