(a) Let ? be a real number. Compute A ? ?I. (b) Find the eigenvalues of...

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(a) Let ? be a real number. Compute A ? ?I.
(b) Find the eigenvalues of A, that is, find the values of ? forwhich the matrix A ? ?I is not invertible. (Hint: There should beexactly 2. Label the larger one ?1 and the smaller ?2.)
(c) Compute the matrices A ? ?1I and A ? ?2I.
(d) Find the eigenspace associated with ?1, that is the set ofall solutions v = v1 v2 to (A ? ?1I)v = 0.
(e) Find the eigenspace associated with ?2 similarly.
Repeat the process to find the eigenvalues and correspondingeigenspaces for
A = [ 0 2 0
2 0 0
1 1 4 ]
(Note that this matrix has three eigenvalues, not 2.)

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4.1 Ratings (553 Votes)

a We have A I3 2 0 2 0 1 1 4 b The characteristic equation of A is detA I3 0 or 342416 0 or 4 2 2 0 Thus the eigenvalues of A are 1 4 22 and 3 2 c The matrix A 1I3 is 4 2 0 2 4 0 1 1 0 The See Answer

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