1. (i) Show that if A is an n x m matrix and if B is an m x n matrix, then A 0 is an eigenvalue of the n x n matrix AB if and only if A is an eigenvalue of the m x m matrix BA. Show that if m = n then the conclusion is true even for A= 0. (ii) Let M be an mxn matrix (m < n) over R. Show that at least one eigenvalue of the n x n matrix MTM is equal to 0. Show that the eigenvalues of the m xm matrix MMT are also eigenvalues of MT M. %3D

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