Chapter 7 - Eigenvalues and Eigenvectors - 7.1 The Eigenvalue/Eigenvector Problem - Problems - Page 445: 42

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Assume that $Av=\lambda v\\ Bv=\alpha v$ a) Take $(AB)v=A(\alpha v)=\alpha (Av)=\alpha \lambda v=(\lambda \alpha) v$ Hence, $v$ is an eigenvalue of the matrix $AB$ b) Take $(A+B)v=A(v)+B(v)=\lambda v +\alpha v=(\alpha +\lambda) v$ Hence, $v$ is an eigenvalue of the matrix $A+B$