educe from the result of Problem 31 that, for ev- y square matrix A, the matrix eA is nonsingular with Show that eAt = I cos 2t + A sin 2t. Apply this fact to find a general solution of x' = Ax, and verify that it is equivalent to the solution found by the eigenvalue method. e-A uppose that Apply Theorem 3 to calculate the matrix exponential eAt each of the matrices in Problems 35 through 40. for -[: :] A = 1 2 3 how that A²" = I and that A²n+1 = A if n is a positive teger. Conclude that 3 4 0 1 4 0 0 35. А — 36. А — 3 1 eAt = I cosh t + A sinh t, 3 5 20 30 37. A = 1 3 38. А — O 10 20 nd apply this fact to find a general solution of x' = Ax. erify that it is equivalent to the general solution found by Le eigenvalue method. uppose that 1 1 3 3 3 0 1 0 0 2 2 4 4 4 3 3 2 4 4 39. А — 40. А — 0 0 2 0 0 0 3 3 4 A = -2 0 0 0 2
educe from the result of Problem 31 that, for ev- y square matrix A, the matrix eA is nonsingular with Show that eAt = I cos 2t + A sin 2t. Apply this fact to find a general solution of x' = Ax, and verify that it is equivalent to the solution found by the eigenvalue method. e-A uppose that Apply Theorem 3 to calculate the matrix exponential eAt each of the matrices in Problems 35 through 40. for -[: :] A = 1 2 3 how that A²" = I and that A²n+1 = A if n is a positive teger. Conclude that 3 4 0 1 4 0 0 35. А — 36. А — 3 1 eAt = I cosh t + A sinh t, 3 5 20 30 37. A = 1 3 38. А — O 10 20 nd apply this fact to find a general solution of x' = Ax. erify that it is equivalent to the general solution found by Le eigenvalue method. uppose that 1 1 3 3 3 0 1 0 0 2 2 4 4 4 3 3 2 4 4 39. А — 40. А — 0 0 2 0 0 0 3 3 4 A = -2 0 0 0 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![5.6 Matrix Exponentials and Linear Systems 361
Show that eAt
find a general solution of x' =
equivalent to the solution found by the eigenvalue method.
= I cos 2t + A sin 2t. Apply this fact to
Ax, and verify that it is
32. Deduce from the result of Problem 31 that, for ev-
ery square matrix A, the matrix ea is nonsingular with
(A)
33. Suppose that
-[: :]
Apply Theorem 3 to calculate the matrix exponential eAt for
each of the matrices in Problems 35 through 40.
A
1
2
3
Show that A2n = I and that A²n+1
integer. Conclude that
= A if n is a positive
3
35. А —
4
36. А —
1
4
3
1
eAt = I cosh t + A sinh t,
3
4
5
20
30
37. A =
1
3
38. А —
10
20
and apply this fact to find a general solution of x' = Ax.
Verify that it is equivalent to the general solution found by
the eigenvalue method.
34. Suppose that
1
5
1
3
3
4
4
4
3
3
2
4
39. А —
40. А —
2
3
2
4
A
-2
2
3
N O O O](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03106fa3-330c-422f-ad56-a6718a5c3a13%2F86d98b52-16e6-47b6-a2ef-45bf98138fee%2Ftluo59n_processed.png&w=3840&q=75)
Transcribed Image Text:5.6 Matrix Exponentials and Linear Systems 361
Show that eAt
find a general solution of x' =
equivalent to the solution found by the eigenvalue method.
= I cos 2t + A sin 2t. Apply this fact to
Ax, and verify that it is
32. Deduce from the result of Problem 31 that, for ev-
ery square matrix A, the matrix ea is nonsingular with
(A)
33. Suppose that
-[: :]
Apply Theorem 3 to calculate the matrix exponential eAt for
each of the matrices in Problems 35 through 40.
A
1
2
3
Show that A2n = I and that A²n+1
integer. Conclude that
= A if n is a positive
3
35. А —
4
36. А —
1
4
3
1
eAt = I cosh t + A sinh t,
3
4
5
20
30
37. A =
1
3
38. А —
10
20
and apply this fact to find a general solution of x' = Ax.
Verify that it is equivalent to the general solution found by
the eigenvalue method.
34. Suppose that
1
5
1
3
3
4
4
4
3
3
2
4
39. А —
40. А —
2
3
2
4
A
-2
2
3
N O O O
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