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(a) Assume that A1 and 12 are distinct eigenvalues of the nxn matrix A. Prove that an nx1 vector x, which is not a null vector, cannot be an eigenvector of both 1, and 12. Hint: Use proof by contradiction. (b) Assume that 21 and 12 are distinct eigenvalues of the nan matrix A. Prove that if x1 is an eigenvector of 1, and xy is an eigenvector of 12, then xjand x, must be linearly independent vectors. That is, prove that any pair of eigenvectors associated with distinct eigenvalues are linearly independent. Hint: Use proof by contradiction. (c) Two nxn matrices A and B are said to be similar if there exists a nonsingular matrix C such that B = C-lAC. Prove that if A and B are similar matrices, they have the same eigenvalues. Hint: If A is an eigenvalue of B, then B - My = 0. (d) Let M=I-Xxxx, where X in an nxk matrix of rank k. Prove that +(M) = n -k. Hint: Use the properties of the trace operator stated in lectures