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Linear Algebra Quiz 4.2
June 20, 2006 1.(20 points)
4 −2
11
(a) Compute the characteristic polynomial of A.
(b) Compute the eigenvalues of A.
(c) Find a basis for R2 consisting of eigenvalues of A.
(d) Find an invertible matrix P and a diagonal matrix D such that P −1 AP = D.
(e) Compute P −1 .
(f) Write A as a product of P, P −1 , and D (in the correct order!).
(g) Compute eA .
(h) Find a basis for the space of solutions to the system of linear diﬀerential equations
Given A = x1 (t) = 4x1 (t) − 2x2 (t)
x2 (t) = x1 (t) + x2 (t) Solution:
(a) Compute: λ2 − 5λ + 6 = (λ − 3)(λ − 2).
(b) Read oﬀ λ = 2, λ = 3.
1
2
(c) Compute v1 =
, v2 =
.
1
1
12
20
(d) Set P =
. Then P −1 AP =
.
11
03
−1 2
(e) P −1 =
.
1 −1
12
20
−1 2
(f) A = P DP −1 =
.
11
03
1 −1
12
e2 0
−1 2
(g) eA = P eD P −1 =
.
3
11
0e
1 −1
(h) Basis for space of solutions is given by the columns of etA =
−e2t + 2e3t 2(e2t − e3t )
−e2t + 2e3t
, in other words, {
2t
3t
2t
3t
−e + e
2e − e
−e2t + e3t
the space of solutions. 1 12
e2t 0
−1 2
=
3t
1 −1
11
0e
2(e2t − e3t )
,
} is a basis for
2e2t − e3t ...
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.
 Spring '10
 Peters

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