le al 35. (a) Show that the eigenvalues of the 2 × 2 matrix |a b A = are the solutions of the quadratic equation X2 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = }(a + d ± V(a – d)² + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) = A, + ^2 and det A = A,d
le al 35. (a) Show that the eigenvalues of the 2 × 2 matrix |a b A = are the solutions of the quadratic equation X2 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = }(a + d ± V(a – d)² + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) = A, + ^2 and det A = A,d
Linear Algebra: A Modern Introduction
4th Edition
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Author:David Poole
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Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 8BEXP
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![35. (a) Show that the eigenvalues of the 2 X 2 matrix
a b
A =
c d]
с а
are the solutions of the quadratic equation
12 – tr(A)A + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
A = {(a + d ± V(a – d)? + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
tr(A) - λι + λ and det A = λιλε
where A, and A2 are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb88a591-bb84-4ad5-a9d9-9dd6e5ab413a%2Fe87477fc-720b-4471-ac3d-4dab07f2b1c2%2Ftx5x52_processed.png&w=3840&q=75)
Transcribed Image Text:35. (a) Show that the eigenvalues of the 2 X 2 matrix
a b
A =
c d]
с а
are the solutions of the quadratic equation
12 – tr(A)A + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
A = {(a + d ± V(a – d)? + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
tr(A) - λι + λ and det A = λιλε
where A, and A2 are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.
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