Let A = B O O C where B and C are square matrices. (a) If is an eigenvalue of B with eigenvector x = (x1, ... , xk) T , show that is also an eigenvalue of A with eigenvector x = (x1, ... , xk, 0, ... , 0)T . (b) If B and C are positive matrices, show that A has a positive real eigenvalue r with the property that || < r for any eigenvalue = r. Show also that the multiplicity of r is at most 2 and that r has a nonnegative eigenvector. (c) If B = C, show that the eigenvalue r in part (b) has multiplicity 2 and possesses a positive eigenvector.

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