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1) Find the characteristic polynomial and the eigenvalues of the following matrices over C. -1 6 a) A 3 11 3 2 2 b) = 1 4 1 -2 -1 2) Let A € Mn(C) and a € C. Using the definition of the determinant given in class, prove that if B is the matrix obtained by multiplying a single row of A by a. then det(B) = det(A). 3) (#14, Section 5.1) For any square matrix A. prove that A and At have the same characteristic polynomial (and hence the same eigenvalues). You may assume that A € Mn(C) or Mn(R). 4) a) If A and B are two similar n X 72 matrices, prove that A and B have the same characteristic polynomial, and hence, the same eigenvalues. b) Show that if T: Cr C" is a linear map, then the eigenvalues of [2% and the eigenvalues of [7] are the same for any two ordered bases 8.7 of C". Conclude that the eigenvalues of a linear transformation are independent of the matrix form given by a choice of basis. 5) Let A € Mn(C) and suppose A is invertible. Note that zero is not an eigenvalue of A. a) Prove that l is an eigenvalue of A if and only if 1/X is an eigenvalue of A-1 b) Prove that A is diagonalizable if and only if A-1 is diagonalizable. 1 6) Let A = @1,1 @1,2 € M2(R). Define a map HA M2(R) - M2(R) by @2,1 @2,2 HA(B) = 01,101,1 @1,261,2 @2,162,1 @2,262,2 for all B b1,1 61.2 = € M2(R). 62,1 62.2 a) Prove that HA is a linear map. 1 1 b) Let A = Find the eigenvalues of HA.

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Linear Algebra Problems
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