Determine whether the matrix [1000] 0410 0041 A= 0004 is diagonalizable. If not, explain why not; if so, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 (you do not have to compute P-¹).

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Determine whether the matrix
1 0 0 0
0 4 1 0
A
0
0
4 1
0004
is diagonalizable. If not, explain why not; if so, find an invertible matrix P and a diagonal matrix D such
that A = PDP-¹ (you do not have to compute P-¹).
=
Transcribed Image Text:Determine whether the matrix 1 0 0 0 0 4 1 0 A 0 0 4 1 0004 is diagonalizable. If not, explain why not; if so, find an invertible matrix P and a diagonal matrix D such that A = PDP-¹ (you do not have to compute P-¹). =
key
No, A is not diagonalizable. A is (upper) triangular, so its eigenvalues are just its diagonal entries: A = 1
and X = 4 (repeated three times). However, the maximum number of linearly independent 4-eigenvectors is
one, not three, since you can check that A - 47 has only one non-pivot column (free variable).
Answer
Transcribed Image Text:key No, A is not diagonalizable. A is (upper) triangular, so its eigenvalues are just its diagonal entries: A = 1 and X = 4 (repeated three times). However, the maximum number of linearly independent 4-eigenvectors is one, not three, since you can check that A - 47 has only one non-pivot column (free variable). Answer
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