Answer to Question #325179 in Algebra for Layla

Question

a) If 𝐴 =

1

2

[

1 1

1 1

], then 𝐴

100 = (

1

2

)

100 [

1 1

1 1

].

b) For any three matrices 𝐴, 𝐵 and 𝐶, if 𝐴𝐵 = 𝐵𝐴 and 𝐵𝐶 = 𝐶𝐵, then 𝐴𝐶 = 𝐶𝐴.

c) If (𝐼 − 2𝐴

𝑇

)

−1 = [

2 1

1 1

], then 𝐴 =

1

2

[

0 1

1 −1

].

d) If 𝐴 and 𝐵 are 4 × 4 nonsingular matrices such that |2𝐴

−1𝐵

2

| = 32, then 

|𝐴| =

|𝐵|

2

2

.

e) Let 𝑆 = {(1,0,0,0), (01,0,0), (2,3,5𝑎, 0), (𝑎, 0,2, 𝑎 − 2)} be set of vectors in

ℝ

4

. The set 𝑆 is linearly independent if and only if 𝑎 ≠ 0.