5.3. EXERCISES 105
Now give the solution in terms of a,b, and c to 1 0 32 3 41 0 2
x
yz
=
abc
.
41. Using the inverse of the matrix, find the solution to the systems 1 0 32 3 41 0 2
x
yz
=
123
,
1 0 32 3 41 0 2
x
yz
=
210
1 0 3
2 3 41 0 2
x
yz
=
101
,
1 0 32 3 41 0 2
x
yz
=
3−1−2
.
Now give the solution in terms of a,b, and c to 1 0 32 3 41 0 2
x
yz
=
abc
.
42. Using the inverse of the matrix, find the solution to the system−1 1
212
12
3 12 − 1
2 − 52
−1 0 0 1−2 − 3
414
94
xyzw
=
abcd
.
43. Show that if A is an n×n invertible matrix and x is a n×1 matrix such that Ax = bfor b an n×1 matrix, then x = A−1b.
44. Prove that if A−1 exists and Ax = 0 then x = 0.
45. Show that if A−1 exists for an n×n matrix, then it is unique. That is, if BA = I andAB = I, then B = A−1.
46. Show that if A is an invertible n×n matrix, then so is AT and(AT)−1
=(A−1
)T.
47. Show (AB)−1 = B−1A−1 by verifying that AB(B−1A−1
)= I and
B−1A−1 (AB) = I.
Hint: Use Problem 45.
48. Show that (ABC)−1 =C−1B−1A−1 by verifying that
(ABC)(C−1B−1A−1)= I
and(C−1B−1A−1
)(ABC) = I. Hint: Use Problem 45.