348 Ordinary Differential Equations
In exercises 17-28 compute the determinant of the given matrix.
- A= (4) 18. B = (-4) 19.
(~ -D - (V2
J3 D
21. (1 ~ i - i )
l+i
- e-A 1 )
3 4-A
- ( sin x cosx) 24. ( e3x e-x )
cosx -sinx^3 e3x -e-x
u
1
(^4) -D
0
G =~ D (
x x2 x3)
27. 1 2x 3x^2
0 2 6x(-A
4- 3 -2-A ,_f)
-1 0 - Let A be any square matrix of size n x n and let I be the same size
identity matrix. Does (A+AI)^2 = A^2 +2AA+A^2 I , where A^2 =AA and
A is any scalar? (HINT: Consider the distributive law for (A+ AI)^2 =
(A+AI)(A+AI) and then the commutative law for scalar multiplication
and multiplication of a matrix by the identity matrix. If you think the
result is not true, give an example which shows equality does not always
hold.)- Let A and B be any two square matrices of the same size. Does
(A+ B)^2 = A^2 + 2AB + B^2? (HINT: Consider the distributive law
for (A+ B)^2 = (A+ B)(A + B) and the commutativity of AB.)
- If x 1 and x2 are both solutions of Ax = 0 (that is, if Ax 1 = 0 and
Ax2 = 0 ), show that y = c1x 1 + c2x 2 is a solution of Ax= 0 for every
choice of the scalars c1 and c2. - If x 1 is a solution of Ax = 0 and x 2 is a solution of Ax = b show that
y = cx1 + x2 is a solution of Ax = b for every choice of the scalar c.
In exercises 33-36 determine whether the given set of vectors is
linearly independent or linearly dependent.
- { (-~)' (-~)} 34. { G), G)}
- { U). m. G)} 36 { ( -D. (-D. m }