# M3 - characteristic polynomials of similar matrices are the...

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Math 375 Midterm test III 1. Let A = 0 0 2 0 2 - 1 0 5 0 0 7 4 1 2 5 3 . (i) Evaluate the determinant of A . (ii) Find the determinants of A T , A - 1 (if A is invertible), A 100 , ( A T ) 100 . 2. Let T : R n R n be a linear transformation. Prove: The number 0 is an eigenvalue if and only if T is not invertible. 3. Let u 1 = 1 2 , u 2 = 2 - 1 . Denote by A the 2 × 2 matrix which satisfies Au 1 = u 1 , Au 2 = 2 u 2 and find an explicit formula for A and the powers A n , for all n = 1 , 2 , 3 , . . . . 4. Let A = 1 2 3 0 1 2 0 0 2 . Describe the eigenvalues and corresponding eigenspaces. Is A diagonalizable? 5. Let A be an n × n matrix and denote by p A the characteristic polynomial of A , defined by p A ( λ ) = det( λI - A ) . (i) Show that if B is an invertible n × n matrix then p A = p BAB - 1 (i.e. the
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Unformatted text preview: characteristic polynomials of similar matrices are the same). (ii) Conclude that the traces of similar matrices are the same. 6. Do one of the following problems. I: Let A be a real symmetric matrix and let v and w be eigenvectors for diﬀerent eigenvalues λ and μ , λ 6 = μ . Show that v and w are orthogonal. II. Suppose that there are n linearly independent vectors in R n which are eigen-vectors for the n × n matrix A . Prove that there is an invertible n × n matrix P so that P-1 AP is a diagonal matrix....
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