396 CHAPTER 18. LINEAR FUNCTIONS
35. Write
x1 + x2 + x3
2x3 + x1 + x2x3 − x1
3x4 + x1
in the form A
x1x2x3x4
where A is an appropriate matrix.
36. Using the inverse of the matrix, find the solution to the systems 1 0 32 3 41 0 2
xyz
=
123
,
1 0 32 3 41 0 2
xyz
=
210
1 0 3
2 3 41 0 2
xyz
=
101
,
1 0 32 3 41 0 2
xyz
=
3−1−2
.
Now give the solution in terms of a,b, and c to 1 0 32 3 41 0 2
xyz
=
abc
.
37. Using the inverse of the matrix, find the solution to the system3 −2 −1 10 1 1 11 −1 −1 01 1 0 1
xyzw
=
abcd
.
38. 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.
39. Prove that if A−1 exists and Ax= 0 then x= 0.
40. 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.
41. Show that if A is an invertible n×n matrix, then so is AT and(AT)−1
=(A−1
)T.
42. Show (AB)−1 = B−1A−1 by verifying that AB(B−1A−1
)= I and
B−1A−1 (AB) = I. Hint: Use Problem 40.
43. 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 40.
44. If A is invertible, show(A2)−1
=(A−1
)2. Hint: Use Problem 40.
45. If A is invertible, show(A−1
)−1= A. Hint: Use Problem 40.
46. Let A and be a real m× n matrix and let x ∈ Rn and y ∈ Rm. Show (Ax,y)Rm =(x,ATy
)Rn where (·, ·)Rk denotes the dot product in Rk. In the notation above, this
would be written as Ax ·y = x·ATy. Use the definition of matrix multiplication todo this.