8.8. EXERCISES 157

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.

49. If A is invertible, show(A2)−1

=(A−1

)2. Hint: Use Problem 45.

50. If A is invertible, show(A−1

)−1= A. Hint: Use Problem 45.

51. 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, Ax ·y = x·ATy.Use the definition of matrix multiplication to do this.

52. Use the result of Problem 51 to verify directly that (AB)T = BT AT without makingany reference to subscripts.

53. Suppose A is an n×n matrix and for each j,

n

∑i=1

∣∣Ai j∣∣< 1

Show that the infinite series ∑∞k=0 Ak converges in the sense that the i jth entry of

the partial sums converge for each i j. Hint: Let R ≡ max j ∑ni=1

∣∣Ai j∣∣ . Thus R < 1.

Show that∣∣∣(A2

)i j

∣∣∣ ≤ R2. Then generalize to show that∣∣∣(Am)i j

∣∣∣ ≤ Rm. Use this to