Begin2.DVI

10-34. Let A=

[

t^3 +t cos 3 t

e^4 t tanh2t

]

and find

(a)

dA

dt (b)

∫

A dt

10-35. Let C(λ) = |A−λI |=det (A−λI ) = 0 denote the characteristic equation

associated with the matrix Ahaving distinct eigenvalues λ 1 ,... , λ n.

(a) Show that C(λ) = (−1)nλn+c 1 λn−^1 +···cn− 1 λ+cn= (−1)n(λ−λ 1 )(λ−λ 2 )···(λ−λn)

(b) Show that cn=λ 1 λ 2 ···λn=|A|=det A

(c) Show Ais singular if any eigenvalue is zero.

(c) Use the fact that the matrix Asatisfies its own characteristic equation and show

A−^1 =

− 1

cn

[

(−1)nAn−^1 +c 1 An−^2 +··· +cn− 1 I

]

10-36.

(a) Show that A

∫t

0

eAt dt +I=eAt

(b) Show that

∫t

0

eAt dt =A−^1

[

eAt −I

]

=

[

eAt −I

]

A−^1

10-37. Verify the following matrix relations for the n×nmatrix A.

(a) sin A=

eiA −e−iA

2 i where i

(^2) =− 1

(b) cos A=e

iA +e−iA

2

(c) sinh A=e

A−e−A

2

(d) cosh A=

eA+e−A

2

(e) sin^2 A+ cos^2 A=I

(f) cosh^2 A−sinh^2 A=I

10-38. Consider the initial-value matrix differential equation

dX (t)

dt

=AX (t) + F(t), X (t 0 ) = C

where Ais a constant matrix.

(a) Left-multiply the given matrix differential equation by the matrix function

e−A(t−t^0 )and show

d

dt

(

e−A(t−t^0 )X

)

=e−A(t−t^0 )F(t)

(b) Integrate both sides of the result from part (a) from t=t 0 to tand show

X=X(t) = eA(t−t^0 )C+eAt

∫t

t 0

e−AξF(ξ)dξ

(c) Show in the special case F= [0], the solution reduces to X=X(t) = eA(t−t^0 )C