1. **Problem Statement:** Given a matrix $A$ with two eigenvalues equal to 1 each, find the eigenvalues of the matrix $5A$. 2. **Formula and Rules:** - If $\lambda$ is an eigenvalue of matrix $A$, then $k\lambda$ is an eigenvalue of matrix $kA$ for any scalar $k$. - This means eigenvalues scale linearly with scalar multiplication of the matrix. 3. **Given:** Two eigenvalues of $A$ are $1$ and $1$. 4. **Find:** Eigenvalues of $5A$. 5. **Solution:** - Let the eigenvalues of $A$ be $\lambda_1 = 1$, $\lambda_2 = 1$, and $\lambda_3 = \lambda_3$ (unknown third eigenvalue). - The eigenvalues of $5A$ are $5\lambda_1$, $5\lambda_2$, and $5\lambda_3$. - Since two eigenvalues of $A$ are 1, the corresponding eigenvalues of $5A$ are $5 \times 1 = 5$ and $5 \times 1 = 5$. 6. **Final answer:** - Two eigenvalues of $5A$ are $\boxed{5}$ and $\boxed{5}$. - The third eigenvalue of $5A$ is $5\lambda_3$, which depends on the third eigenvalue of $A$ (not given).