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Question 1 Find all square roots of the matrix 5 2 . 3 0 That is to say find all matrices B such that B² - A. Hint: Finding the square root of a diagonal matrix is easy. You can leave the answer as a product. Question 2 Use the Cauchy-Schwarz inequality to prove that for 0 < a < 1 we have that /2 - 1 J0 Question 1 True or false: (a) Every linear operator in an n-dimensional vector space has n distinct eigenvalues. (b) There exists a square real matrix with no real eigenvalues. (c) There exists a square matrix with no (complex) eigenvalues. (d) Similar matrices always have the same eigenvalues. (e) Similar matrices always have the same eigenvectors. (f) The sum of two eigenvectors of a matrix is A is also an eigenvector of A. Question 2 Find the characteristic polynomials, eigenvalues and eigenvectors of the following matrices: 1 3 3 (==) 4 2 3 , ( - 2 1 4 1 3 - 5 -3 , . 3 3 1 Question 3 Recall that a linear operator A is nilpotent if Ak: = 0 for some integer k > 1. Prove that if A is nilpotent, then o(A)=(0). = Question 4 Let A be an n X n matrix. True or false: (a) AT has the same eigenvalues as A. (b) AT has the same eigenvectors as A. (c) If A is diagonalisable, then so is AT. Question 5 Suppose that A is a square real matrix. Show that if \ € C \ R is an eigenvalue of A with eigenvector U € Cn, then I is an eigenvalue of A with eigenvector U. Question 6 Construct a matrix A with eigenvalues 1 and 3 with corresponding eigen- vectors (1,2) and (1,1), respectively. Is this matrix unique? Question 7 Diagonalise the following matrices if possible: 4 1 - 1 2 ). , 6 4 1 . Question 8 Consider the matrix 2 6 -6 0 5 -2 0 0 4 (a) Find the eigenvalues of A. Is it possible to find them without comput- ing? (b) Argue, without computing anything, that A is diagonalisable. (c) Diagonalise A. Question 9 Find all square roots of the matrix 5 2 . -3 0 That is to say find all matrices B such that B² - A. Hint: Finding the square root of a diagonal matrix is easy. You can leave the answer as a product.

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